{"id":860,"date":"2025-06-20T17:18:34","date_gmt":"2025-06-20T17:18:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=860"},"modified":"2025-09-09T19:44:58","modified_gmt":"2025-09-09T19:44:58","slug":"basics-of-differential-equations-learn-it-2-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/basics-of-differential-equations-learn-it-2-2\/","title":{"raw":"Sequences and Their Properties: Learn It 2","rendered":"Sequences and Their Properties: Learn It 2"},"content":{"raw":"

Limit of a Sequence<\/h2>\r\n

Now that we understand what sequences are, let's explore a fundamental question: What happens to the terms of a sequence as [latex]n[\/latex] gets larger and larger?<\/p>\r\n

Since sequences are functions defined on positive integers, we can discuss what happens to [latex]a_n[\/latex] as [latex]n \\to \\infty[\/latex]. Let's examine four different sequences to see the various behaviors that can occur.<\/p>\r\n\r\n

\r\n

Example a:<\/strong> [latex]{1+3n} = {4, 7, 10, 13, \\ldots}[\/latex]<\/p>\r\n

The terms [latex]1+3n[\/latex] grow without bound as [latex]n \\to \\infty[\/latex]. We say [latex]1+3n \\to \\infty[\/latex] as [latex]n \\to \\infty[\/latex].<\/p>\r\n

Example b:<\/strong> [latex]{1-(\\frac{1}{2})^n} = {\\frac{1}{2}, \\frac{3}{4}, \\frac{7}{8}, \\frac{15}{16}, \\ldots}[\/latex]<\/p>\r\n

The terms get closer and closer to 1. We say [latex]1-(\\frac{1}{2})^n \\to 1[\/latex] as [latex]n \\to \\infty[\/latex].<\/p>\r\n

Example c:<\/strong> [latex]{(-1)^n} = {-1, 1, -1, 1, \\ldots}[\/latex]<\/p>\r\n

The terms alternate between -1 and 1 forever. They don't settle down to any single value.<\/p>\r\n

Example d:<\/strong> [latex]{\\frac{(-1)^n}{n}} = {-1, \\frac{1}{2}, -\\frac{1}{3}, \\frac{1}{4}, \\ldots}[\/latex]<\/p>\r\n

The terms alternate in sign but get closer and closer to 0. We say [latex]\\frac{(-1)^n}{n} \\to 0[\/latex] as [latex]n \\to \\infty[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"721\"]\"Four Figure 2. (a) The terms in the sequence become arbitrarily large as [latex]n\\to \\infty [\/latex]. (b) The terms in the sequence approach [latex]1[\/latex] as [latex]n\\to \\infty [\/latex]. (c) The terms in the sequence alternate between [latex]1[\/latex] and [latex]-1[\/latex] as [latex]n\\to \\infty [\/latex]. (d) The terms in the sequence alternate between positive and negative values but approach [latex]0[\/latex] as [latex]n\\to \\infty [\/latex].[\/caption]<\/section>From these examples, we see that sequences can behave in different ways as [latex]n[\/latex] gets large. In Examples b and c, the terms approach a specific finite number. In Examples a and c, they don't. If the terms of a sequence approach a finite number [latex]L[\/latex] as [latex]n\\to \\infty [\/latex], we say that the sequence is a convergent sequence and the real number [latex]L[\/latex] is the limit of the sequence.\r\n\r\n

\r\n

convergent and divergent sequences<\/h3>\r\n

A sequence [latex]{a_n}[\/latex] is convergent<\/strong> if the terms [latex]a_n[\/latex] get arbitrarily close to some finite number [latex]L[\/latex] as [latex]n[\/latex] becomes sufficiently large.<\/p>\r\n

We write:<\/p>\r\n

[latex]\\lim_{n\\to \\infty}a_n = L[\/latex]<\/p>\r\n

The number [latex]L[\/latex] is called the limit of the sequence<\/strong>.<\/p>\r\n

If a sequence is not convergent, we say it is divergent<\/strong>.<\/p>\r\n\r\n<\/section>

\r\n

Convergent vs. Divergent<\/strong><\/p>\r\n\r\n

    \r\n \t
  • Convergent:<\/strong> The terms \"settle down\" to approach a specific finite value<\/li>\r\n \t
  • Divergent:<\/strong> The terms either grow without bound, oscillate forever, or behave erratically<\/li>\r\n<\/ul>\r\n

    Remember: Even if terms alternate (like in Example 4), a sequence can still converge if the alternating terms get closer to a single value.<\/p>\r\n\r\n<\/section>Looking at our examples more closely, we can see that [latex]{1-(\\frac{1}{2})^n}[\/latex] has terms that get arbitrarily close to 1 as [latex]n[\/latex] becomes very large. This makes it a convergent sequence with limit 1. In contrast, [latex]{1+3n}[\/latex] has terms that keep growing without approaching any finite number, making it divergent.\r\n

    The Formal Definition<\/h3>\r\n

    Our informal description used phrases like \"arbitrarily close\" and \"sufficiently large,\" which are helpful but somewhat vague. Here's the precise mathematical definition and show these ideas graphically in Figure 3.<\/p>\r\n\r\n

    \r\n

    formal definition of sequence convergence<\/h3>\r\n

    A sequence [latex]{a_n}[\/latex] converges<\/strong> to a real number [latex]L[\/latex] if:<\/p>\r\n

    For every [latex]\\epsilon > 0[\/latex], there exists an integer [latex]N[\/latex] such that [latex]|a_n - L| < \\epsilon[\/latex] whenever [latex]n \\geq N[\/latex].<\/p>\r\n

    We write:<\/p>\r\n

    [latex]\\underset{n\\to \\infty }{\\text{lim}}a_n = L[\/latex] or [latex]a_n \\to L[\/latex]<\/p>\r\n

    If a sequence doesn't converge, it's divergent<\/strong>.<\/p>\r\n\r\n<\/section>

    What this means in plain English:<\/strong> No matter how small a distance [latex]\\epsilon[\/latex] you choose around [latex]L[\/latex], eventually all the terms of the sequence will be within that distance of [latex]L[\/latex].<\/section>[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"A Figure 3. As [latex]n[\/latex] increases, the terms [latex]{a}_{n}[\/latex] become closer to [latex]L[\/latex]. For values of [latex]n\\ge N[\/latex], the distance between each point [latex]\\left(n,{a}_{n}\\right)[\/latex] and the line [latex]y=L[\/latex] is less than [latex]\\epsilon [\/latex].[\/caption]\r\n

    The convergence of a sequence depends only on what happens as [latex]n \\to \\infty[\/latex]. This means you can add or remove any finite number of terms from the beginning of a sequence without changing whether it converges or diverges.<\/p>\r\n\r\n

    For example, if [latex]{a_n}[\/latex] converges to [latex]L[\/latex], then the sequence [latex]{b_1, b_2, \\ldots, b_N, a_1, a_2, \\ldots}[\/latex] will also converge to [latex]L[\/latex], regardless of what the values [latex]b_1, b_2, \\ldots, b_N[\/latex] are.<\/section>\r\n

    Not all divergent sequences behave the same way. Let's look at the sequences [latex]{1+3n}[\/latex] and [latex]{(-1)^n}[\/latex] from our earlier examples to see two distinct types of divergence.<\/p>\r\n

    Oscillating divergence:<\/strong> The sequence [latex]{(-1)^n} = {-1, 1, -1, 1, \\ldots}[\/latex] diverges because terms alternate between 1 and -1 forever, never settling on a single value.<\/p>\r\n

    Divergence to infinity:<\/strong> The sequence [latex]{1+3n}[\/latex] diverges because the terms grow without bound: [latex]1+3n \\to \\infty[\/latex] as [latex]n \\to \\infty[\/latex].<\/p>\r\n

    For sequences that grow without bound, we write [latex]\\lim_{n\\to \\infty}(1+3n) = \\infty[\/latex]. Similarly, sequences can diverge to negative infinity<\/strong>. For example, [latex]{-5n+2}[\/latex] has terms that approach [latex]-\\infty[\/latex], so we write [latex]\\lim_{n\\to \\infty}(-5n+2) = -\\infty[\/latex].<\/p>\r\n\r\n

    \r\n

    Important Note About Infinity<\/strong><\/p>\r\n

    When we write [latex]\\lim_{n\\to \\infty}a_n = \\infty[\/latex], we're not<\/strong> saying the limit exists. The sequence is still divergent! This notation just tells us how<\/strong> it diverges\u2014by growing without bound rather than oscillating.<\/p>\r\n\r\n<\/section>\r\n

    Using Function Limits to Find Sequence Limits<\/h3>\r\n

    Since sequences are functions defined on positive integers, we can often use our knowledge of function limits to analyze sequence behavior.<\/p>\r\n

    Here's the key insight: If you have a sequence [latex]{a_n}[\/latex] where [latex]a_n = f(n)[\/latex] for some function [latex]f[\/latex], and if [latex]\\lim_{x\\to \\infty}f(x) = L[\/latex], then the sequence converges to the same limit [latex]L[\/latex].<\/p>\r\n\r\n

    Consider the sequence [latex]{\\frac{1}{n}}[\/latex] and the related function [latex]f(x) = \\frac{1}{x}[\/latex]. Since [latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{x} = 0[\/latex], we know that [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{1}{n} = 0[\/latex].<\/section>
    \r\n

    limit of a sequence defined by a function<\/h3>\r\n

    Consider a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] such that [latex]{a}_{n}=f\\left(n\\right)[\/latex] for all [latex]n\\ge 1[\/latex]. If there exists a real number [latex]L[\/latex] such that<\/p>\r\n\r\n

    [latex]\\underset{x\\to \\infty }{\\text{lim}}f\\left(x\\right)=L[\/latex],<\/div>\r\n

    then [latex]\\left\\{{a}_{n}\\right\\}[\/latex] converges and<\/p>\r\n\r\n

    [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}=L[\/latex].<\/div>\r\n<\/section>
    Why this works:<\/strong> The sequence values are just a subset of the function values (at integer points), so if the function approaches [latex]L[\/latex], the sequence must too.<\/section>
    \r\n

    This method is especially useful when:<\/p>\r\n\r\n

      \r\n \t
    • Your sequence formula looks like a familiar function<\/li>\r\n \t
    • You can easily find the limit of the continuous version<\/li>\r\n \t
    • The function techniques (like L'H\u00f4pital's rule) are simpler than working directly with the sequence<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"

      Limit of a Sequence<\/h2>\n

      Now that we understand what sequences are, let’s explore a fundamental question: What happens to the terms of a sequence as [latex]n[\/latex] gets larger and larger?<\/p>\n

      Since sequences are functions defined on positive integers, we can discuss what happens to [latex]a_n[\/latex] as [latex]n \\to \\infty[\/latex]. Let’s examine four different sequences to see the various behaviors that can occur.<\/p>\n

      \n

      Example a:<\/strong> [latex]{1+3n} = {4, 7, 10, 13, \\ldots}[\/latex]<\/p>\n

      The terms [latex]1+3n[\/latex] grow without bound as [latex]n \\to \\infty[\/latex]. We say [latex]1+3n \\to \\infty[\/latex] as [latex]n \\to \\infty[\/latex].<\/p>\n

      Example b:<\/strong> [latex]{1-(\\frac{1}{2})^n} = {\\frac{1}{2}, \\frac{3}{4}, \\frac{7}{8}, \\frac{15}{16}, \\ldots}[\/latex]<\/p>\n

      The terms get closer and closer to 1. We say [latex]1-(\\frac{1}{2})^n \\to 1[\/latex] as [latex]n \\to \\infty[\/latex].<\/p>\n

      Example c:<\/strong> [latex]{(-1)^n} = {-1, 1, -1, 1, \\ldots}[\/latex]<\/p>\n

      The terms alternate between -1 and 1 forever. They don’t settle down to any single value.<\/p>\n

      Example d:<\/strong> [latex]{\\frac{(-1)^n}{n}} = {-1, \\frac{1}{2}, -\\frac{1}{3}, \\frac{1}{4}, \\ldots}[\/latex]<\/p>\n

      The terms alternate in sign but get closer and closer to 0. We say [latex]\\frac{(-1)^n}{n} \\to 0[\/latex] as [latex]n \\to \\infty[\/latex].<\/p>\n

      \"Four
      Figure 2. (a) The terms in the sequence become arbitrarily large as [latex]n\\to \\infty [\/latex]. (b) The terms in the sequence approach [latex]1[\/latex] as [latex]n\\to \\infty [\/latex]. (c) The terms in the sequence alternate between [latex]1[\/latex] and [latex]-1[\/latex] as [latex]n\\to \\infty [\/latex]. (d) The terms in the sequence alternate between positive and negative values but approach [latex]0[\/latex] as [latex]n\\to \\infty [\/latex].<\/figcaption><\/figure>\n<\/section>\n

      From these examples, we see that sequences can behave in different ways as [latex]n[\/latex] gets large. In Examples b and c, the terms approach a specific finite number. In Examples a and c, they don’t. If the terms of a sequence approach a finite number [latex]L[\/latex] as [latex]n\\to \\infty[\/latex], we say that the sequence is a convergent sequence and the real number [latex]L[\/latex] is the limit of the sequence.<\/p>\n

      \n

      convergent and divergent sequences<\/h3>\n

      A sequence [latex]{a_n}[\/latex] is convergent<\/strong> if the terms [latex]a_n[\/latex] get arbitrarily close to some finite number [latex]L[\/latex] as [latex]n[\/latex] becomes sufficiently large.<\/p>\n

      We write:<\/p>\n

      [latex]\\lim_{n\\to \\infty}a_n = L[\/latex]<\/p>\n

      The number [latex]L[\/latex] is called the limit of the sequence<\/strong>.<\/p>\n

      If a sequence is not convergent, we say it is divergent<\/strong>.<\/p>\n<\/section>\n

      \n

      Convergent vs. Divergent<\/strong><\/p>\n

        \n
      • Convergent:<\/strong> The terms “settle down” to approach a specific finite value<\/li>\n
      • Divergent:<\/strong> The terms either grow without bound, oscillate forever, or behave erratically<\/li>\n<\/ul>\n

        Remember: Even if terms alternate (like in Example 4), a sequence can still converge if the alternating terms get closer to a single value.<\/p>\n<\/section>\n

        Looking at our examples more closely, we can see that [latex]{1-(\\frac{1}{2})^n}[\/latex] has terms that get arbitrarily close to 1 as [latex]n[\/latex] becomes very large. This makes it a convergent sequence with limit 1. In contrast, [latex]{1+3n}[\/latex] has terms that keep growing without approaching any finite number, making it divergent.<\/p>\n

        The Formal Definition<\/h3>\n

        Our informal description used phrases like “arbitrarily close” and “sufficiently large,” which are helpful but somewhat vague. Here’s the precise mathematical definition and show these ideas graphically in Figure 3.<\/p>\n

        \n

        formal definition of sequence convergence<\/h3>\n

        A sequence [latex]{a_n}[\/latex] converges<\/strong> to a real number [latex]L[\/latex] if:<\/p>\n

        For every [latex]\\epsilon > 0[\/latex], there exists an integer [latex]N[\/latex] such that [latex]|a_n - L| < \\epsilon[\/latex] whenever [latex]n \\geq N[\/latex].<\/p>\n

        We write:<\/p>\n

        [latex]\\underset{n\\to \\infty }{\\text{lim}}a_n = L[\/latex] or [latex]a_n \\to L[\/latex]<\/p>\n

        If a sequence doesn’t converge, it’s divergent<\/strong>.<\/p>\n<\/section>\n

        What this means in plain English:<\/strong> No matter how small a distance [latex]\\epsilon[\/latex] you choose around [latex]L[\/latex], eventually all the terms of the sequence will be within that distance of [latex]L[\/latex].<\/section>\n
        \"A
        Figure 3. As [latex]n[\/latex] increases, the terms [latex]{a}_{n}[\/latex] become closer to [latex]L[\/latex]. For values of [latex]n\\ge N[\/latex], the distance between each point [latex]\\left(n,{a}_{n}\\right)[\/latex] and the line [latex]y=L[\/latex] is less than [latex]\\epsilon [\/latex].<\/figcaption><\/figure>\n

        The convergence of a sequence depends only on what happens as [latex]n \\to \\infty[\/latex]. This means you can add or remove any finite number of terms from the beginning of a sequence without changing whether it converges or diverges.<\/p>\n

        For example, if [latex]{a_n}[\/latex] converges to [latex]L[\/latex], then the sequence [latex]{b_1, b_2, \\ldots, b_N, a_1, a_2, \\ldots}[\/latex] will also converge to [latex]L[\/latex], regardless of what the values [latex]b_1, b_2, \\ldots, b_N[\/latex] are.<\/section>\n

        Not all divergent sequences behave the same way. Let’s look at the sequences [latex]{1+3n}[\/latex] and [latex]{(-1)^n}[\/latex] from our earlier examples to see two distinct types of divergence.<\/p>\n

        Oscillating divergence:<\/strong> The sequence [latex]{(-1)^n} = {-1, 1, -1, 1, \\ldots}[\/latex] diverges because terms alternate between 1 and -1 forever, never settling on a single value.<\/p>\n

        Divergence to infinity:<\/strong> The sequence [latex]{1+3n}[\/latex] diverges because the terms grow without bound: [latex]1+3n \\to \\infty[\/latex] as [latex]n \\to \\infty[\/latex].<\/p>\n

        For sequences that grow without bound, we write [latex]\\lim_{n\\to \\infty}(1+3n) = \\infty[\/latex]. Similarly, sequences can diverge to negative infinity<\/strong>. For example, [latex]{-5n+2}[\/latex] has terms that approach [latex]-\\infty[\/latex], so we write [latex]\\lim_{n\\to \\infty}(-5n+2) = -\\infty[\/latex].<\/p>\n

        \n

        Important Note About Infinity<\/strong><\/p>\n

        When we write [latex]\\lim_{n\\to \\infty}a_n = \\infty[\/latex], we’re not<\/strong> saying the limit exists. The sequence is still divergent! This notation just tells us how<\/strong> it diverges\u2014by growing without bound rather than oscillating.<\/p>\n<\/section>\n

        Using Function Limits to Find Sequence Limits<\/h3>\n

        Since sequences are functions defined on positive integers, we can often use our knowledge of function limits to analyze sequence behavior.<\/p>\n

        Here’s the key insight: If you have a sequence [latex]{a_n}[\/latex] where [latex]a_n = f(n)[\/latex] for some function [latex]f[\/latex], and if [latex]\\lim_{x\\to \\infty}f(x) = L[\/latex], then the sequence converges to the same limit [latex]L[\/latex].<\/p>\n

        Consider the sequence [latex]{\\frac{1}{n}}[\/latex] and the related function [latex]f(x) = \\frac{1}{x}[\/latex]. Since [latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{x} = 0[\/latex], we know that [latex]\\underset{n\\to \\infty }{\\text{lim}}\\frac{1}{n} = 0[\/latex].<\/section>\n
        \n

        limit of a sequence defined by a function<\/h3>\n

        Consider a sequence [latex]\\left\\{{a}_{n}\\right\\}[\/latex] such that [latex]{a}_{n}=f\\left(n\\right)[\/latex] for all [latex]n\\ge 1[\/latex]. If there exists a real number [latex]L[\/latex] such that<\/p>\n

        [latex]\\underset{x\\to \\infty }{\\text{lim}}f\\left(x\\right)=L[\/latex],<\/div>\n

        then [latex]\\left\\{{a}_{n}\\right\\}[\/latex] converges and<\/p>\n

        [latex]\\underset{n\\to \\infty }{\\text{lim}}{a}_{n}=L[\/latex].<\/div>\n<\/section>\n
        Why this works:<\/strong> The sequence values are just a subset of the function values (at integer points), so if the function approaches [latex]L[\/latex], the sequence must too.<\/section>\n
        \n

        This method is especially useful when:<\/p>\n

          \n
        • Your sequence formula looks like a familiar function<\/li>\n
        • You can easily find the limit of the continuous version<\/li>\n
        • The function techniques (like L’H\u00f4pital’s rule) are simpler than working directly with the sequence<\/li>\n<\/ul>\n<\/section>\n","protected":false},"author":15,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":671,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/860"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/860\/revisions"}],"predecessor-version":[{"id":2259,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/860\/revisions\/2259"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/671"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/860\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=860"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=860"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=860"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}