Subject: Fwd: convergence

Hi, I am a high school senior and I need help stugying for a final. I am stuck on one of the questions on my review sheet. Does the improper integral from 5 to infinity of (38/97)^x converge or diverge? If it converges I also need to know how to find the approximate value accurate to .01 of its actual value.
Thanks

In your calculus class you have probably spent quite a bit of time on THE exponential function e^x and not much time on other exponential functions like 7^x or (38/97)^x. The reason is that any exponential function a^x can be written with base e as follows:

Suppose that y = a^x

then ln(y) = ln(a^x) = x ln(a)

and hence y = e^ln(y) = e^{x ln(a)}.

In your problem (38/97)^x = e^{x ln(38/79)} and hence find

recalling that ln(38/79) = ln(38) - ln(79).