Section 5.3 Calculus 2

Series

5.3 Series

5.3.1 Series as Partial Sum Sequences

• For a given sequence $$\<a_n\>_{n=0}^\infty$$, its partial sum sequence $$\<s_n\>_{n=0}^\infty$$ is defined explicitly by $$s_n=\sum_{i=0}^n a_i=a_0+a_1+\dots+a_n$$, and defined recursively by $$s_0=a_0$$ and $$s_{n+1}=s_n+a_{n+1}$$.
• Example Write out the first few terms of the partial sum sequence for $$\<1,2,3,4,5,\dots\>$$.
• Example Write out the first few terms of the partial sum sequence for $$\<b_i\>_{i=1}^\infty$$ where $$b_i=\frac{6}{i}$$.
• The series $$\sum_{n=0}^\infty a_n=a_0+a_1+a_2+\dots$$ represents the sum of the infinite sequence $$\<a_n\>_{n=0}^\infty$$. If its partial sum sequence converges to $$L$$, then we say that its series converges to $$L$$ and the value of the series is $$L$$ (written $$\sum_{n=0}^\infty a_n=a_0+a_1+a_2+\dots=L$$). Otherwise, we say the series diverges.

5.3.2 Finding Limits of Partial Sum Sequences

• A telescoping series is a series whose partial sum sequence allows for canceling.
• Example Show that $$\sum_{n=1}^\infty(\frac{1}{n}-\frac{1}{n+1})$$ converges to $$1$$ by evaluating the limit of its partial sum sequence.
• Example Does $$\sum_{n=0}^\infty\frac{2}{n^2+3n+2}$$ converge or diverge?

(TODO record video)

• The geometric series defined for real numbers $$a,r$$ is $$\sum_{n=0}^\infty ar^n=a+ar+ar^2+ar^3+\dots$$.
• The limit of a geometric partial sum sequence $$s_n=\sum_{k=0}^\infty ar^k$$ may be computed by first finding $$s_n-rs_n=a-ar^{n+1}$$ and then dividing by $$r$$. This limit only exists when $$|r|<1$$.
• Example Find the value of $$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$$.
• Example Does $$\sum_{k=1}^\infty \frac{4^n}{3^{n+2}}$$ converge or diverge?

5.3.3 Divergent Series

• The Series Divergence Test: If a sequence fails to converge to $$0$$, then its series diverges.
• Example Does $$\sum_{k=0}^\infty\frac{k^2+3}{2k^2+k+5}$$ converge or diverge? If it converges, what is its value?
• This does NOT mean that if a sequence converges, then its series converges.
• The harmonic sequence $$\<\frac{1}{n}\>_{n=1}^\infty$$ converges to $$0$$, but its series $$\sum_{n=1}^\infty\frac{1}{n}$$ diverges.

5.3.4 Arithmetic Rules and Reindexing

• Because a series is a limit, it follows the same rules as limits do.
• Example Evaluate the convergent series $$\sum_{i=0}^\infty\frac{1+\frac{2^{i+2}}{i+1}-\frac{2^{i+2}}{i+2}}{2^i}$$.
• The starting index for a series may be adjusted by offsetting the index for its sequence in the opposite direction.
• Example Does $$\sum_{m=-1}^\infty\frac{1}{m+2}$$ converge or diverge? If it converges, what is its value?

Review Exercises

1. Write out the first four terms of the partial sum sequence for $$\<1,-\frac{1}{3},\frac{1}{9},-\frac{1}{27},\dots\>$$.
2. Write out the first four terms of the partial sum sequence for $$\<0.3,0.03,0.003,0.0003,\dots\>$$.
3. Does $$\sum_{m=2}^\infty(\frac{3}{2m}-\frac{3}{2m+2})$$ converge or diverge? If it converges, what is its value?
4. Does $$\sum_{j=2}^\infty\frac{6}{4j^2+4j}$$ converge or diverge? If it converges, what is its value?
5. Compute $$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots$$.
6. Prove that $$0.\overline3=0.333\dots$$ equals $$\frac{1}{3}$$ by expressing the decimal expression as a geometric series.
7. Write $$0.\overline{27}=0.272727\dots$$ as a fraction of integers.
8. Does $$\sum_{n=0}^\infty\frac{6}{3^{n+2}}$$ converge or diverge? If it converges, what is its value?
9. Does $$\sum_{m=0}^\infty 3(-1)^m$$ converge or diverge? If it converges, what is its value?
10. Does $$\sum_{i=1}^\infty \frac{i+\sin i}{2i}$$ converge or diverge? If it converges, what is its value?
11. Suppose $$\sum_{n=0}^\infty a_n=3$$ and $$\sum_{n=0}^\infty b_n=4$$. Evaluate $$\sum_{n=0}^\infty(3a_n-2b_n)$$.
12. Does $$\sum_{k=2}^\infty 4(\frac{2}{3})^k$$ converge or diverge? If it converges, what is its value?
13. Prove $$\sum_{n=1}^\infty\frac{1}{3^n}=\frac{1}{2}$$ using the proof of the Geometric Series formula (not the formula itself).
14. Does $$\sum_{n=3}^\infty\left(\frac{6}{n}-\frac{6}{n+1}\right)$$ converge or diverge? If it converges, what is its value?
15. Does $$\sum_{i=0}^\infty\frac{(-3)^i}{2}$$ converge or diverge? If it converges, what is its value?
16. Does $$\sum_{n=1}^\infty\frac{1}{4^n}$$ converge or diverge? If it converges, what is its value?

Solutions 1-7

Solutions 8-16

Textbook References

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