Please help with the following questions:

Convergence Tests Test for Divergence. If lim a, does not exist or if lim an * 0 then the series ) is divergent. n=1 Remark. If lim an = 0, the test is inconclusive. This test cannot prove the convergence of a series. n + 10 1. 100n + 1 2. sinn 1=1 The Integral Test. Suppose f is a function that is positive, continuous, and decreasing on [1, co). Let an = f(n), then the series ) on is convergent If and only if the improper integral / f(x) dr is convergent. Remark. . When we use the Integral Test, it is not necessary to start the series or the integral at n = 1. For example, in testing the series _ _3) we use ) (2 3, de . Also, it is not necessary that f be always decreasing. What is important is that f be ultimately decreasing, that is, decreasing for r larger than some number N. Then the series _ on is convergent if and only if the improper integral f(x) dx is convergent. Since a finite number of terms doesn't affect the convergence or divergence of a series, ) a, is convergent if and only if the improper integral / f(x) dr is convergent.