Series Convergence Tests

Add up infinitely many shrinking numbers, and sometimes you land on a finite total. The running total, the partial sum $S_N = \sum_{n=1}^{N} a_n$, is what actually converges (or not). Geometric $\sum 1/2^n$ climbs to 1; the Basel sum $\sum 1/n^2$ to $\pi^2/6$; the alternating $\sum (-1)^{n+1}/n$ zigzags onto $\ln 2$. But the harmonic sum $\sum 1/n$ never settles: it crawls past every bound even though its terms go to zero. That is the crucial lesson, terms shrinking to zero is necessary but not sufficient. The scene runs the partial sum toward its limit; the diagnostic plots the term sizes $|a_n|$ on a log scale, so the decay rate that decides convergence is visible.

Figure 1. Series convergence. Top: the partial sum S_N as N grows, approaching the limit for a convergent series or climbing without bound for the harmonic series. Bottom: the term magnitudes |a_n| on a log axis, the decay rate that decides convergence. Method: direct partial summation.

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exponent p0.70

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WHAT TO TRY

Take the p-series and drag the exponent across p = 1: below it the sum climbs forever (diverges), above it the sum settles onto a limit. The verdict flips exactly at the boundary the p-test predicts.
Switch to geometric and slide the ratio r toward ±1: the sum stays finite while |r| < 1 and blows up at the edge, the ratio test live.
Switch to the alternating series and drop p below 1: it still converges (conditionally), the partial sums bracketing the limit ever more tightly even though the same terms summed in absolute value would diverge.
Watch the term-size plot: geometric terms fall on a straight line (exponential decay), power-law terms fall off far more slowly, which is why the boundary sits where it does.