Since the function is non-negative and decreasing on [1,∞) we can use the integral test. The integral converges by the *p*-test . The integral test says that the series must also converge. In pictures, the area described by the integral is finite. Since the rectangles that correspond to the terms of the series cover a smaller area, this smaller area must also be finite. This means the series converges. There's something a tiny bit tricky going on in this example. The rectangles that give the underestimate of don't correspond exactly to the series They correspond to the series because of how we're drawing the rectangles: This is one of those times when limits of summation don't matter . The first term *a*_{1} of the series is a finite number, so if converges then so does the original series This is why we have the part about *c* in the integral test. Since limits of summation don't matter for the convergence/divergence of a series, if converges, then so does We can use any starting index we like, so long as all the terms are defined. This means the convergence/divergence of the infinite series *a*_{1} + ... + *a*_{c – 1 + }*a*_{c + ...}
is determined by the converge/divergence of its *tail* *a*_{c} + *a*_{c + 1} + .... If the tail converges, the whole series converges. If the tail diverges, the series diverges. This means if the starting index of the series isn't a good choice for *c*, we can pick a different value. We can pick any whole number value of *c* for which *f* is non-negative and decreasing on [*c*,∞). |