Sure, let's examine this step-by-step:
0. Fill in the blanks.
- A sequence is a list of numbers.
- A series is a sum of numbers. A series converges if its partial sum has a limit.
- The nth term test tells us that the series [tex]\(\sum_{n=1}^{\infty} a_n\)[/tex] diverges if [tex]\(\lim_{n \to \infty} a_n \neq\)[/tex] 0 and if [tex]\(\lim_{n \to \infty} a_n =\)[/tex] 0 it tells us nothing conclusive about the series' convergence.
1. Determine whether the series [tex]\(\sum_{n=5}^{\infty}\left(1+\frac{1}{n}\right)^n\)[/tex] converges or diverges. If it converges, find the sum.
To determine whether the series converges or diverges, we will use the nth term test.
The nth term [tex]\( a_n \)[/tex] of the series is given by [tex]\( a_n = \left(1 + \frac{1}{n}\right)^n \)[/tex].
First, we need to find the limit of [tex]\( a_n \)[/tex] as [tex]\( n \)[/tex] approaches infinity:
[tex]\[
\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n
\][/tex]
It is known that:
[tex]\[
\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e
\][/tex]
where [tex]\( e \approx 2.718281828459045 \)[/tex].
So,
[tex]\(\lim_{n \to \infty} \left(1+\frac{1}{n}\right)^n = 2.718281828459045\)[/tex].
According to the nth term test, if the limit of [tex]\( a_n \)[/tex] as [tex]\( n \)[/tex] approaches infinity is not zero, the series diverges.
Since [tex]\( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.718281828459045 \neq 0 \)[/tex],
the series [tex]\(\sum_{n=5}^{\infty} \left(1 + \frac{1}{n}\right)^n\)[/tex] diverges.
