Infinite Sequences and Series
Core ideas
Sequences have limits; series are sums of sequences. Convergence tests form a toolkit: nth-term test (diverges if terms do not go to zero), geometric series ($\sum ar^n$ converges to $\frac{a}{1-r}$ for $|r|<1$), p-series (converges for $p>1$), comparison and limit comparison, integral test, ratio and root tests, and alternating series test with error bound. Absolute versus conditional convergence matters.
Power series and Taylor series represent functions near a point: $\sum\frac{f^{(n)}(a)}{n!}(x-a)^n$. Know the standard Maclaurin series for $e^x$, $\sin x$, $\cos x$, and $\frac{1}{1-x}$, and the radius of convergence via the ratio test. Error estimates come from the alternating series bound or Taylor remainder.
Traps
- The nth-term test proves divergence only; terms going to zero does not imply convergence (harmonic series).
- Mixing up the ratio test's inconclusive case ($L=1$) with divergence.
- Forgetting the geometric convergence condition $|r|<1$.
- Endpoint behavior of an interval of convergence must be checked separately.
Worked example
$\sum_{n=1}^{\infty}\frac{1}{2^n}$ is geometric with $a=\frac{1}{2}$, $r=\frac{1}{2}$, so the sum is $\frac{1/2}{1-1/2}=1$. By contrast, the harmonic series $\sum\frac{1}{n}$ diverges even though its terms tend to zero.