The Main Idea 

A power series is essentially a polynomial that never stops growing. Instead of having a finite number of terms like [latex]3 + 2x + x^2[/latex], a power series continues with infinitely many powers of [latex]x[/latex].

The basic form: [latex]\displaystyle\sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots[/latex]

Here, [latex]c_n[/latex] represents the coefficients (just numbers), and [latex]x[/latex] is your variable. This is called a power series centered at zero because the powers are of [latex]x[/latex] itself.

Centered at any point: You can also have [latex]\displaystyle\sum_{n=0}^{\infty} c_n (x-a)^n[/latex], which is centered at [latex]x = a[/latex]. This shifts the “center” of your series from zero to the point [latex]a[/latex].

Unlike regular series with just numbers, power series involve a variable [latex]x[/latex]. This means convergence depends on what value you plug in for [latex]x[/latex]. The series might converge for some values of [latex]x[/latex] and diverge for others.

Power series bridge the gap between polynomials (which you can easily work with) and more complex functions. They let you represent functions like [latex]e^x[/latex], [latex]\sin x[/latex], and [latex]\cos x[/latex] as infinite sums, opening up powerful computational and analytical techniques.

The big question with any power series is: “For which values of [latex]x[/latex] does this infinite sum actually converge to a finite value?”

The Main Idea 

The critical question for any power series is: “For which values of [latex]x[/latex] does this infinite sum actually give you a finite answer?” Unlike regular series with constant terms, power series convergence depends entirely on what you plug in for [latex]x[/latex].

One guaranteed convergence point: Every power series [latex]\displaystyle\sum_{n=0}^{\infty} c_n (x-a)^n[/latex] always converges at its center [latex]x = a[/latex]. When you substitute [latex]x = a[/latex], all terms except the first become zero, leaving just [latex]c_0[/latex].

Three possible behaviors: Every power series falls into exactly one of these categories:

• Case 1: Converges only at the center – The series works at [latex]x = a[/latex] but diverges everywhere else. This happens with series that grow very rapidly, like [latex]\sum n! x^n[/latex].
• Case 2: Converges everywhere – The series converges for all real numbers [latex]x[/latex]. Examples include [latex]\sum \frac{x^n}{n!}[/latex] (which represents [latex]e^x[/latex]).
• Case 3: Converges in an interval – This is the most common case. There exists a positive number [latex]R[/latex] (called the radius of convergence) such that:
  • The series converges when [latex]|x-a| < R[/latex]
  • The series diverges when [latex]|x-a| > R[/latex]
  • At the boundary points [latex]|x-a| = R[/latex], you need to check individually

The endpoints [latex]x = a-R[/latex] and [latex]x = a+R[/latex] require separate investigation. The series might converge at one endpoint, both endpoints, or neither – there’s no universal rule.

The Main Idea 

Once you know a power series converges in an interval, you need to find exactly where that interval is and how wide it extends. The interval of convergence tells you all the [latex]x[/latex]-values where your series actually works, while the radius of convergence [latex]R[/latex] measures how far you can go from the center before the series breaks down.

For a power series [latex]\displaystyle\sum_{n=0}^{\infty} c_n (x-a)^n[/latex], the radius [latex]R[/latex] represents the distance from the center [latex]a[/latex] to the edge of convergence:

• [latex]R = 0[/latex]: Series only works at [latex]x = a[/latex]
• [latex]R = \infty[/latex]: Series works everywhere
• [latex]R > 0[/latex]: Series works when [latex]|x-a| < R[/latex] and fails when [latex]|x-a| > R[/latex]

Finding the radius with the ratio test: Apply the ratio test to get [latex]\rho = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|[/latex]. For convergence, you need [latex]\rho < 1[/latex], which typically gives you a condition like [latex]|x-a| < R[/latex].

When [latex]|x-a| = R[/latex] (at [latex]x = a-R[/latex] and [latex]x = a+R[/latex]), the ratio test gives [latex]\rho = 1[/latex], making it inconclusive. You must test these boundary points separately by substituting them into the original series and checking convergence using other tests.

Common endpoint patterns:

• Alternating harmonic-type series often converge at one endpoint
• Regular harmonic-type series usually diverge at endpoints
• Check each endpoint individually – they can behave differently

The interval of convergence might be [latex](a-R, a+R)[/latex], [latex][a-R, a+R)[/latex], [latex](a-R, a+R][/latex], or [latex][a-R, a+R][/latex], depending on endpoint behavior.

The Main Idea 

Power series give you a way to rewrite complicated functions as “infinite polynomials,” which are much easier to work with. Once you have a power series representation, you can differentiate, integrate, and evaluate the function term by term – operations that might be difficult or impossible with the original form.

The key insight comes from the geometric series formula. For [latex]|r| < 1[/latex], we have [latex]\frac{a}{1-r} = a + ar + ar^2 + ar^3 + \cdots[/latex]. This gives us our most important power series representation:

[latex]\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots[/latex] for [latex]|x| < 1[/latex]

Most functions won’t look exactly like [latex]\frac{1}{1-r}[/latex], but you can often manipulate them algebraically to fit this pattern. The key is recognizing forms that can be rewritten as geometric series.

Common algebraic tricks:

• Rewrite [latex]\frac{1}{1+x^3}[/latex] as [latex]\frac{1}{1-(-x^3)}[/latex] to use [latex]r = -x^3[/latex]
• Factor constants from denominators: [latex]\frac{x^2}{4-x^2} = \frac{x^2}{4(1-\frac{x^2}{4})}[/latex]
• Look for patterns that match [latex]\frac{a}{1-r}[/latex] where [latex]r[/latex] involves your variable

Power series representations let you:

• Approximate function values using partial sums (just add up the first few terms)
• Integrate or differentiate functions that would be difficult in their original form
• Analyze function behavior near specific points

Your power series representation is only valid within its interval of convergence. Outside this interval, the infinite sum doesn’t equal your original function, so always identify where [latex]|r| < 1[/latex] for your specific [latex]r[/latex].