Lecture 7 Series Convergence & Integral Test

#series #convergence #divergence

This lecture focuses on determining whether an infinite sum (series) $\sum_{n = 1}^{\infty} a_{n}$ converges (reaches a finite sum) or diverges (goes to infinity or oscillates).

1. Telescoping Series 🔭

A series is "telescoping" if the terms cancel each other out, leaving only the first and last parts.

Structure: $a_{n} = b_{n} - b_{n + 1}$
Partial Sum Formula ( $s_{n}$ ): $s_{n} = b_{1} - b_{n + 1}$
Infinite Sum ( $S$ ): $S = lim_{n \to \infty} (b_{1} - b_{n + 1})$

**⚠️ Always write out the first 3 terms to see the cancellation pattern before jumping to the formula!

2. The Divergence Test (The "Check This First" Test) 🛑

This is the fastest test. Use it before doing any hard math.

The Rule: If $lim_{n \to \infty} a_{n} \neq 0$ , then the series must diverge.
The Trap: If $lim_{n \to \infty} a_{n} = 0$ , the test is inconclusive. The series might converge, or it might still diverge (like the Harmonic Series $\sum \frac{1}{n}$ ).

3. The Integral Test ♾️

We compare the series to an improper integral.

Conditions for use:

The function $f (n) = a_{n}$ must be:

Positive
Continuous
Decreasing (on the interval $[N, \infty)$ )

The Theorem:

The series $\sum a_{n}$ and the integral $\int_{N}^{\infty} f (x) d x$ behave the same way.

If the integral is a number $⟹$ Converges.
If the integral is $\infty ⟹$ Diverges.

4. The p-Series Shortcut ⚡

A specialized result of the Integral Test for series in the form $\sum_{n = 1}^{\infty} \frac{1}{n^{p}}$ .

Value of $p$	Result	Example
$p > 1$	Convergent	$\sum \frac{1}{n^{2}}$
$p \leq 1$	Divergent	$\sum \frac{1}{\sqrt{n}}$ or $\sum \frac{1}{n}$

5. Estimating Sums (Remainders) 📉

Since we can't sum to infinity, we use the Nth partial sum ( $s_{N}$ ) and estimate the Remainder ( $R_{N}$ ).

The Remainder Bound:

The "leftover" error $R_{N}$ is trapped between two integrals:

\int_{N + 1}^{\infty} f (x) d x \leq R_{N} \leq \int_{N}^{\infty} f (x) d x

Finding $N$ for a specific error:

To ensure the error is less than a value (e.g., $0.001$ ), solve this inequality for $N$ :

\int_{N}^{\infty} f (x) d x < Error Tolerance

💡 Quick Strategy Guide

Divergence Test: Do the terms go to zero? If not, stop—it diverges.
Special Form? Is it a p-series or a Geometric series? Use the shortcut.
Telescoping? Can you write it as $(b_{n} - b_{n + 1})$ ? Find the partial sum.
Integral Test: If none of the above work, can you integrate the function easily? If yes, do the improper integral.