Series Convergence

Definitions

A series is $\sum_{k=0}^{\infty} u_k$. The partial sums are $S_n = \sum_{k=0}^{n} u_k$.

Convergence: The series converges if $\lim_{n \to \infty} S_n = S$ exists (finite). We write $\sum u_k = S$.

Formal definition: For any $\varepsilon > 0$, there exists $N$ such that $|S - S_n| < \varepsilon$ for all $n > N$.

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Necessary Condition for Convergence

If $\sum u_k$ converges, then $\lim_{k \to \infty} u_k = 0$.

Warning: The converse is false. Example: harmonic series $\sum \frac{1}{k}$ diverges even though $\frac{1}{k} \to 0$.

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Absolute vs Conditional Convergence

Absolute convergence: $\sum |u_k|$ converges.

Conditional convergence: $\sum u_k$ converges but $\sum |u_k|$ diverges.

Key theorem: If $\sum |u_k|$ converges, then $\sum u_k$ converges.

Riemann rearrangement theorem: Conditionally convergent series can be rearranged to converge to any value (or diverge). Absolutely convergent series are immune to rearrangement.

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Comparison Test

For series with $0 \leq u_k \leq v_k$:

1. If $\sum v_k$ converges, then $\sum u_k$ converges
2. If $\sum u_k$ diverges, then $\sum v_k$ diverges

Limit comparison: If $\lim_{k \to \infty} \frac{u_k}{v_k} = L$ with $0 < L < \infty$, then $\sum u_k$ and $\sum v_k$ behave the same.

Example: $\sum \frac{1}{k(k+1)}$ converges since $\frac{1}{k(k+1)} < \frac{1}{k^2}$ and $\sum \frac{1}{k^2}$ converges.

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Ratio Test

Let $L = \lim_{k \to \infty} \left|\frac{u_{k+1}}{u_k}\right|$:

• $L < 1$: series converges absolutely
• $L > 1$: series diverges
• $L = 1$: test is inconclusive

Example: $\sum \frac{k^2}{2^k}$. Ratio: $\frac{(k+1)^2/2^{k+1}}{k^2/2^k} = \frac{1}{2}\left(1 + \frac{1}{k}\right)^2 \to \frac{1}{2} < 1$. Converges.

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Harmonic Series Divergence

$\sum_{k=1}^{\infty} \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$

Proof of divergence (grouping):

$1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \cdots$

$> 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots$

The partial sums grow without bound.

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Alternating Series (Leibniz Criterion)

An alternating series $\sum (-1)^k a_k$ with $a_k > 0$ converges if:

1. $a_k$ is monotonically decreasing for large $k$
2. $\lim_{k \to \infty} a_k = 0$

Example: The alternating harmonic series: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2$

This is conditionally convergent (not absolutely, since harmonic series diverges).

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Riemann Zeta Function

$\zeta(p) = \sum_{k=1}^{\infty} \frac{1}{k^p}$

• Converges for $p > 1$
• Diverges for $p \leq 1$

Special values:

• $\zeta(2) = \frac{\pi^2}{6}$
• $\zeta(4) = \frac{\pi^4}{90}$

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Radius of Convergence for Power Series

For $\sum a_n x^n$, the radius of convergence is:

$R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right| \quad \text{(if limit exists)}$

• Converges absolutely for $|x| < R$
• Diverges for $|x| > R$
• At $x = \pm R$: must check separately

Example: $\sum \frac{x^n}{n}$.

$R = \lim \frac{1/n}{1/(n+1)} = 1$. Converges for $|x| < 1$.

At $x = 1$: harmonic series diverges. At $x = -1$: alternating harmonic converges.

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Summary: Choosing a Test

Situation	Preferred Test
Comparison with known series	Comparison test
Factorials, exponentials	Ratio test
Alternating signs	Leibniz criterion
Powers $1/k^p$	p-test (zeta)
Power series	Radius of convergence

Complexity: Ratio test is often easiest to apply. Comparison test requires knowing a suitable reference series.