Power series let us write many functions as infinite polynomials, which are often easier to approximate, differentiate, integrate, and analyze. Building new series means starting from a familiar series and transforming it into a series for a new function. This matters because one known pattern, such as the geometric series, can generate many useful expansions.
The main tools are substitution, multiplication, term-by-term differentiation, and term-by-term integration.
The most important starting point is 1/(1 - x) = sum from n = 0 to infinity of x^n, valid for |x| < 1. By replacing x with another expression, multiplying by powers or constants, or differentiating and integrating each term, you can create new power series with predictable coefficients. The radius and interval of convergence must be checked after every transformation, especially at endpoints.
A worked example often follows the path from a known input series through an operation, then to a simplified output series.
Understanding Calculus: Building New Series
A useful way to think about a series transformation is to track two things at every step. Track the function on the left and track the general term on the right. For example, replacing the input by x squared makes every power even.
The resulting terms have powers zero, two, four, and so on. Replacing the input by negative x makes signs alternate from term to term. Replacing the input by three x makes the coefficients grow by powers of three.
These changes are not cosmetic. They determine which values of x make the series behave properly.
Substitution requires more care than it first appears to need. The original series has a condition on its input, but after substitution the new input is a whole expression. Students must solve the condition using that expression before they state an interval.
If the new input is two x minus one, the allowed x values are found by requiring the size of two x minus one to be less than one. This can produce an interval with a different center from zero. A series can therefore be centered at a value other than zero, even when it came from a familiar pattern centered at zero.
Multiplication creates another common source of errors. Multiplying a series by a single power of x is straightforward because every term shifts by the same amount. The coefficients stay in the same order.
Multiplying two full series is different. Terms with the same final power must be collected together. For the coefficient of a given power, add every product of earlier coefficients whose powers combine to make that power.
This process is called a convolution. It appears in signal processing, probability, and physics because combined effects often come from many smaller pairs of contributions.
When working by hand, write out several terms before trying to state a general pattern. This prevents missed terms and incorrect indices.
Differentiation and integration change more than the appearance of powers. Differentiation can remove a constant term and increases the weight of higher powers through their exponents. Integration adds a new constant that cannot be recovered from the series alone.
This constant comes from information about the original function, such as a known value at one point. Endpoint checks deserve their own line of work. A transformed series may work throughout its open interval yet fail at one or both boundary values.
Substitute each endpoint into the actual series and use an appropriate convergence test. Alternating signs, comparison with a known series, or telescoping patterns can settle the result. Clear index handling, careful conditions, and separate endpoint tests are the habits that make built series reliable.
Key Facts
- Geometric series: 1/(1 - x) = sum from n = 0 to infinity of x^n, valid for |x| < 1.
- Substitution rule: replace x by g(x) to get 1/(1 - g(x)) = sum from n = 0 to infinity of (g(x))^n, valid when |g(x)| < 1.
- Term-by-term differentiation: if f(x) = sum a_n x^n, then f'(x) = sum from n = 1 to infinity of n a_n x^(n - 1) inside the radius of convergence.
- Term-by-term integration: if f(x) = sum a_n x^n, then integral f(x) dx = C + sum from n = 0 to infinity of a_n x^(n + 1)/(n + 1) inside the radius of convergence.
- Multiplying by x^k shifts powers: x^k sum a_n x^n = sum a_n x^(n + k).
- Endpoint convergence must be tested separately because differentiation and integration keep the same radius of convergence but may change endpoint behavior.
Vocabulary
- Power series
- A power series is an infinite sum of the form sum a_n (x - c)^n, where c is the center and a_n are coefficients.
- Geometric series
- A geometric series is a series with a constant ratio between consecutive terms, such as sum x^n = 1/(1 - x) for |x| < 1.
- Radius of convergence
- The radius of convergence is the distance from the center within which a power series is guaranteed to converge.
- Term-by-term differentiation
- Term-by-term differentiation means differentiating each term of a power series to create a series for the derivative.
- Interval of convergence
- The interval of convergence is the set of x-values where a power series converges, including any endpoints that pass separate tests.
Common Mistakes to Avoid
- Forgetting to update the convergence condition after substitution is wrong because |x| < 1 becomes a new condition such as |2x| < 1 or |x^2| < 1.
- Differentiating the function but not the series terms correctly is wrong because each term a_n x^n becomes n a_n x^(n - 1), and the n = 0 term disappears.
- Assuming endpoints are automatically included is wrong because the radius gives an open interval first, and each endpoint must be tested with the resulting numerical series.
- Losing the constant of integration is wrong because an antiderivative series should include C unless a specific initial value determines it.
Practice Questions
- 1 Use 1/(1 - x) = sum from n = 0 to infinity of x^n to find a power series for 1/(1 + 3x). State the interval of convergence.
- 2 Starting with 1/(1 - x), differentiate term by term to find a power series for 1/(1 - x)^2. Then write the first four nonzero terms.
- 3 A student replaces x with x^2 in the geometric series and says the interval of convergence is still -1 < x < 1 because the original series used |x| < 1. Explain the correct reasoning and whether the final interval changes.