Calculus Worksheets: Integrating Power Series Term by Term Exercises

📚 Topic Summary

Integrating power series term by term allows us to find the antiderivative of a function represented as a power series. This involves applying the power rule for integration to each term of the series. The resulting series represents the integral of the original function, with an added constant of integration, $C$. This technique is particularly useful when the function doesn't have an elementary antiderivative.

When integrating power series, it's important to remember that the interval of convergence might change. After integrating, you should re-evaluate the interval of convergence for the new power series.

🧠 Part A: Vocabulary

Match the term with its definition:

1. Term: Power Series
2. Term: Interval of Convergence
3. Term: Radius of Convergence
4. Term: Antiderivative
5. Term: Term-by-Term Integration

Definitions:

1. A function $F(x)$ such that $F'(x) = f(x)$.
2. The set of all $x$ values for which the power series converges.
3. A series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$.
4. The radius $R$ such that the power series converges for $|x-a| < R$.
5. Integrating each term of a power series individually.

✍️ Part B: Fill in the Blanks

When integrating a power series term by term, we apply the ______ rule to each term. The interval of ______ may change after integration. We must also remember to add the constant of ______, denoted by $C$. This method is useful when a function doesn't have an ______ antiderivative.

Answer: power, convergence, integration, elementary

🤔 Part C: Critical Thinking

Explain why term-by-term integration is a useful technique when dealing with functions that do not have elementary antiderivatives. Provide an example of such a function and explain how power series integration can be applied.