How do you know when a series converges

For a series $\sum_{n=1}^{\infty}a_n$ the partial sum $S_k=\sum_{n=1}^{k}a_n$ forms a new sequence. If $\lim_{k\to\infty}S_k=L$ exists and is a real number, the series converges to $L$. Otherwise the series diverges. This definition turns the problem of an infinite sum into a limit problem that calculus tools can handle.

Common Convergence Tests

The most frequently used tests are:

• nth‑term test (divergence test)
• Geometric series test
• p‑series test
• Comparison test
• Limit comparison test
• Ratio test
• Root test
• Integral test

Choose a test that matches the form of $a_n$. If the series looks like a power of a constant, the geometric test is quickest. Polynomial denominators suggest a p‑series or comparison. Exponential factors often yield a ratio or root test. When the terms involve an integral‑like expression, the integral test can be decisive.

Follow these steps to decide convergence:

1. 1Write down $a_n$ and check $\lim_{n\to\infty}a_n$; if it is not zero, the series diverges (nth‑term test).
2. 2Identify a familiar pattern (geometric, p‑series, alternating) and apply the corresponding test.
3. 3If no pattern is obvious, use the ratio or root test because they work for factorials and exponentials.
4. 4When the ratio/root test is inconclusive (limit =1), switch to comparison, limit comparison, or integral test.

Worked example: determine convergence of $\sum_{n=1}^{\infty}\frac{2^n}{3^n}$. Simplify the term: $a_n=(2/3)^n$. This is a geometric series with ratio $r=2/3$. Since $|r|<1$, the series converges and its sum is $\frac{r}{1-r}=\frac{2/3}{1-2/3}=2$. The ratio test also confirms convergence because $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\frac{2}{3}<1$.

Comparison of typical series:

Remember that convergence requires the whole tail of partial sums to settle, not merely the individual terms. After applying a test, verify any conditions (positivity, monotonicity) the test assumes. Practicing each test on concrete series builds intuition for picking the right tool under exam pressure.