# Section 5.4 Calculus 2

Geometric and Alternating Series

## 5.4 Geometric and Alternating Series

### 5.4.1 Geometric Series Test

• The geometric series $$\sum_{n=0}^\infty ar^n$$ converges to $$\frac{a}{1-r}$$ when $$|r|<1$$, and diverges when $$|r|\geq 1$$.
• Example Compute $$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$$.
• Example Does $$\sum_{k=0}^\infty\frac{2}{3^{k+1}}$$ converge or diverge? If it converges, what is its value?
• Example Does $$\sum_{k=0}^\infty\frac{2}{(1/3)^{k+1}}$$ converge or diverge? If it converges, what is its value?

(TODO record)

• Example Prove that $$0.\overline4=0.444\dots$$ equals $$\frac{4}{9}$$ by expressing the decimal expression as a geometric series.

### 5.4.2 Alternating Series Test

• Let $$\<a_n\>_{n=0}^\infty$$ be a monotonic sequence of positive terms. Then $$\sum_{n=0}^\infty(-1)^n a_n=a_0-a_1+a_2-a_3+\dots$$ is called an alternating series.
• An alternating series $$\sum_{n=0}^\infty(-1)^n a_n$$ converges when $$\lim_{n\to\infty} a_n = 0$$, and diverges otherwise.
• This test cannot be used to tell what value the alternating series converges to.
• This test also holds for $$\sum_{n=N}^\infty(-1)^{n\pm k} a_n$$, as long as the terms $$a_n$$ are positive and monotonic.
• Example Show that the alternating harmonic series $$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n} =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots$$ converges.
• Example Does the series $$\sum_{n=0}^\infty\cos(n\pi)(3+\frac{1}{n^2+1})$$ converge or diverge?
• Example Does the series $$\sum_{m=2}^\infty(-1)^m\frac{m}{m^{3/2} +3}$$ converge or diverge?

### Review Exercises

1. Compute $$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots$$.
2. Prove that $$0.\overline3=0.333\dots$$ equals $$\frac{1}{3}$$ by expressing the decimal expression as a geometric series.
3. Write $$5.\overline{27}=5.272727\dots$$ as a fraction of integers.
4. Does $$\sum_{n=0}^\infty\frac{6}{3^{n+2}}$$ converge or diverge? If it converges, what is its value?
5. Does $$\sum_{m=0}^\infty 3(-1)^m$$ converge or diverge? If it converges, what is its value?
6. Prove $$\sum_{n=1}^\infty\frac{1}{3^n}=\frac{1}{2}$$ by mimicking the proof of the Geometric Series formula.
7. Does $$\sum_{n=0}^\infty(-1)^{n+1}\frac{4}{n^2+3}$$ converge or diverge?
8. Does $$\sum_{i=6}^\infty(-1)^i\frac{i}{\sqrt{i^3-7}}$$ converge or diverge?
9. Does $$\sum_{m=2}^\infty(-\frac{3}{5})^m$$ converge or diverge?
10. Does $$\sum_{k=2}^\infty(-\frac{5}{3})^k$$ converge or diverge?
11. Does $$\sum_{n=13}^\infty(-1)^n\frac{1}{n\ln n}$$ converge or diverge?

Solutions 1-6

Solutions 7-11

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.2, 9.6