Series
Does a series converge, and how — absolutely or conditionally; the sum of a geometric series, Taylor coefficients and the radius of convergence of a power series.
How to solve it
A series is an infinite sum. It converges if the partial sums tend to a number. The first question is whether the terms go to zero: if not, the series certainly diverges. If they do, a test decides: comparison with a known series, the ratio of neighbouring terms, alternating signs.
Step by step
- The necessary condition: if $a_n \not\to 0$, the series diverges.
- Factorials and powers: the ratio test. If the limit of $\left|\frac{a_{n+1}}{a_n}\right|$ is below $1$ it converges, above $1$ it diverges.
- Fractions with powers of $n$: compare with $\sum \frac{1}{n^p}$, which converges for $p > 1$.
- An alternating series converges absolutely if the series of absolute values converges; if not, but the absolute values decrease to zero, it converges conditionally (Leibniz's test).
- A geometric series is summed by the formula; take Taylor coefficients from the known series of $e^x$, $\sin x$, $\ln(1 + x)$, $\frac{1}{1 - x}$.
Common mistakes
- Concluding convergence from $a_n \to 0$: the harmonic series $\sum \frac{1}{n}$ diverges.
- Drawing a conclusion at $q = 1$ in the ratio test, where it is silent.
- Mixing up absolute and conditional convergence: $\sum \frac{(-1)^n}{n}$ converges conditionally, $\sum \frac{(-1)^n}{n^3}$ absolutely.
- Taking $b_0$ instead of $b_1$ as the first term of a geometric series that starts at $n = 1$.