However, the upper step function is the lower one shifted one unit to the right. Consequently, the series converge or diverge as the integral does. Note that all three converge, however they are not equal. Another important test is the Limit Comparison Test. A simplified version of this test asserts that if the ratio of the terms of two different sequences have a finite, non-zero limit, then both converge or both diverge.
Lets examine why test works from a geometric point of view.
Consider these three sequences. For now we are not sure if ak converges or diverges. The red graph is subject ak, while bk is blue and ck is green. We now see that things are changing. Ultimately, ak and bk are more or less in sync.