Question

How do you prove that in Calculus that the Sum of (n-5)/(n8^n) from n=5 to infinity is convergent

Answer 1

is it
\[\sum\frac{n-5}{n^n}\]?

Answer 2

\[\sum_{n=5}^{\infty} \frac{ n-5 }{ n8^{n}}\]

Answer 3

Split it into two sums:
\[
\frac{n}{n\cdot8^n}-\frac{5}{n\cdot8^n}=\frac{1}{1\cdot8^n}-\frac{5}{n\cdot8^n}
\]Note that:
\[
\frac{5}{n\cdot8^n}<\frac{5}{8^n}
\]And
\[
\frac{5}{8^n}
\]Converges, hence:
\[
\frac{5}{n\cdot8^n}
\]Also converges.

Since this is the difference of convergent series, the result must converge.

Answer 4

\(\dfrac{n-5}{n}\left(\dfrac{1}{8}\right)^{n} < \dfrac{n}{n}\left(\dfrac{1}{8}\right)^{n} = \left(\dfrac{1}{8}\right)^{n}\)

The one on the right is a Geometric Series and is known to converge.

Answer 5

THANK YOU! BOTH