An infinite series is the sum of an infinite sequence: $\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \dots$. But how can we add infinitely many numbers? We define the sum as the limit of the partial sums $S_N = \sum_{n=1}^N a_n$ as $N \to \infty$. If this limit exists (finite), the series converges; otherwise, it diverges.
🎯 In this section you will learn
- The definition of an infinite series and partial sums
- The difference between convergent and divergent series
- How to find the sum of a telescoping series
- The harmonic series and why it diverges
📌 Definition of an Infinite Series
$$ \sum_{n=1}^\infty a_n = \lim_{N \to \infty} S_N \quad \text{where} \quad S_N = \sum_{n=1}^N a_n $$
If the limit exists (finite), the series converges; otherwise, it diverges
💡 Key Idea
The sum of an infinite series is the limit of its partial sums. This is similar to improper integrals: we take a limit to define the infinite sum.
Consider $\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$
①
Compute partial sums
$S_1 = \frac{1}{2} = 0.5$
$S_2 = \frac{1}{2} + \frac{1}{4} = 0.75$
$S_3 = 0.875$
$S_4 = 0.9375$
$S_5 = 0.96875$
②
Observe the pattern
$S_N = 1 - \frac{1}{2^N}$
③
Take the limit
$\lim_{N \to \infty} (1 - \frac{1}{2^N}) = 1$
④
Conclusion
The series converges to $1$.
Find the sum of $\sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n+1}\right)$.
①
Write the partial sum
$S_N = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \dots + \left(\frac{1}{N} - \frac{1}{N+1}\right)$
②
Cancel terms (telescoping)
$S_N = 1 - \frac{1}{N+1}$
③
Take the limit
$\lim_{N \to \infty} \left(1 - \frac{1}{N+1}\right) = 1$
Consider $\sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$
- Partial sums: $S_1=1$, $S_2=1.5$, $S_4 \approx 2.083$, $S_8 \approx 2.718$, $S_{16} \approx 3.381$
- The partial sums grow slowly but without bound.
- $\lim_{N \to \infty} S_N = \infty$ → the harmonic series diverges.
Show that $\sum_{n=1}^\infty \frac{n}{n+1}$ diverges.
- $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0$
- If $\lim_{n \to \infty} a_n \neq 0$, the series diverges (nth Term Test).
📌 Key Terms
| Term | Definition |
|---|
| $n$th term | $a_n$, the $n$th term of the series |
| Partial sum $S_N$ | $\sum_{n=1}^N a_n$, the sum of the first $N$ terms |
| Convergent series | $\lim_{N \to \infty} S_N = L$ (finite) |
| Divergent series | $\lim_{N \to \infty} S_N$ does not exist or is infinite |
| Telescoping series | Series where terms cancel when writing partial sums |
⚠️ nth Term Test for Divergence
If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.
Note: If the limit is $0$, the series may converge or diverge (inconclusive).
🔍 Key Takeaways
- A series converges if its partial sums approach a finite limit.
- Telescoping series often have simple partial sums that allow easy limit calculation.
- The harmonic series $\sum 1/n$ diverges (important to memorize).
- The nth Term Test can only prove divergence (not convergence).
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