# Section 5.8 Calculus 2

Strategies to Determine Series Convergence

## 5.8 Strategies to Determine Series Convergence

### 5.8.1 Identifying Appropriate Strategies for Inspecting Series

• When encountering a series, it’s useful to spot certain traits that can identify the best series convergence test to apply. The following list isn’t fool-proof, but checking these in order can help you identify useful techniques to determine series convergence.
1. Is the value of a convergent series asked for? If so, it is likely a telescoping or geometric series.
2. Is the sequence limit of its terms non-zero? If so, it diverges by the Series Divergence Test. (The opposite is NOT true.)
3. Is the series of the form $$\sum ar^n$$? If so, it is a geometric series.
4. Do the series terms alternate due to a multiplier of $$(-1)^n$$ or similar? If so, it is likely an alternating series.
5. Is the series of the form $$\sum\frac{a}{n^p}$$? If so, it is a $$p$$-series.
6. Does the series involve factorials? If so, the Ratio Test may be effective (but fails when the limit equals $$1$$).
7. Does the series involve exponents? If so, the Root Test may be effective (but fails when the limit equals $$1$$).
8. Can the series be slightly modified to a geometric or $$p$$-series? If so, then either the Direct or Limit Comparison Test may be effective.
• Note that the starting index (the bottom of the $$\sum$$) does not affect whether a series converges or diverges (but it does affect the value of a convergent series).

### Review Exercises

1. Does $$\sum_{m=1}^\infty(-1)^{m-1}\frac{m+1}{2m}$$ converge or diverge?
2. Does $$\sum_{n=0}^\infty\frac{n+3}{3^n}$$ converge or diverge?
3. Does $$\sum_{k=0}^\infty 7^{1-k}$$ converge or diverge? If it converges, what is its value?
4. Does $$\sum_{j=4}^\infty\frac{5^j}{(2j)!}$$ converge or diverge?
5. Does $$\sum_{n=3}^\infty\frac{1}{n^2+n}$$ converge or diverge? If it converges, what is its value?
6. Does $$\sum_{k=1}^\infty(1+\frac{1}{k})^k$$ converge or diverge?
7. Does $$\sum_{m=2}^\infty5(-\frac{4}{9})^{m+1}$$ converge or diverge?
8. Does $$\sum_{n=0}^\infty\frac{n^3+n+7}{n^5+5n^2}$$ converge or diverge?
9. Does $$\sum_{k=10}^\infty\frac{4}{\sqrt{k}}$$ converge or diverge?
10. For what values of $$x$$ does $$\sum_{n=0}^\infty \frac{x^n}{n^2}$$ converge?