Calculus II: Mastering Series and Sequences

Geometric Series

A geometric series is a specific type of series where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$). The general form is: $\sum_{n=1}^{\infty} a_1 r^{n-1}$

The behavior of a geometric series depends entirely on $r$:

1. If $|r| < 1$, the series converges.
2. If $|r| \geq 1$, the series diverges.

If the series converges, you can calculate the sum of the series using the formula: $S = \frac{a_1}{1 - r}$

For example, in the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{2^{n+1} 3^{n-1}}$, you must first identify the first term $a_1$ and the common ratio $r$ to determine if it converges and what its sum is.

Telescoping Series and Partial Sums

A telescoping series is a series where most terms cancel each other out when you write out the sum. To analyze these, we look at the sequence of partial sums ($S_n$). The $n$-th partial sum is the sum of the first $n$ terms of the series.

If the limit of the sequence of partial sums exists ($\lim_{n \to \infty} S_n = L$), then the series converges to $L$.

A classic example is the series $\sum_{n=1}^{\infty} (\arctan(n) - \arctan(n+1))$. By writing out the terms:

• $S_1 = \arctan(1) - \arctan(2)$
• $S_2 = (\arctan(1) - \arctan(2)) + (\arctan(2) - \arctan(3)) = \arctan(1) - \arctan(3)$
• $S_n = \arctan(1) - \arctan(n+1)$

As $n \to \infty$, $\arctan(n+1) \to \frac{\pi}{2}$. Thus, the sum is $\arctan(1) - \frac{\pi}{2}$.

Comparison Tests

When a series doesn't fit a standard pattern like geometric or telescoping, we use comparison tests to compare it to a series we already know (like a $p$-series or a geometric series).

Direct Comparison Test (DCT)

If you have two series $\sum a_n$ and $\sum b_n$ with positive terms:

• If $a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.
• If $a_n \geq b_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Limit Comparison Test (LCT)

If the direct comparison is difficult due to messy inequalities, use the Limit Comparison Test. If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $0 < c < \infty$, then both series behave the same way (both converge or both diverge).

Example Practice: To test the convergence of $\sum_{n=1}^{\infty} \frac{n+1}{\sqrt{n^3}}$, we compare it to the $p$-series $\sum \frac{n}{n^{3/2}} = \sum \frac{1}{n^{1/2}}$. Since $\sum \frac{1}{\sqrt{n}}$ is a divergent $p$-series ($p = 1/2 \leq 1$), the original series also diverges.

Summary Table for Series Testing

r| < 1$. | | Telescoping | Terms cancel out (often via partial fractions) | Converges if $\lim S\_n$exists. | | Comparison | Series "looks like" a$p$-series or geometric | Depends on the comparison series. |