Question

Select the true statement about the series, ¾ + 7/8 + 11/12 + 15/16…
a. The series is given by n/n+1, where n is a natural number.
b.The series is n + 4/s +4 times the sequence, where n and s are natural numbers.
c.The series is divergent.

Answer 1

It can't be the first statement, since
\[\sum_{n=1}^\infty \frac{n}{n+1}=\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots\]
The second statement also isn't right, because you can pick any two natural numbers that would be completely different from the given series. For example, if both \(n\) and \(s\) start at 1, then each term would be \(\dfrac{n+4}{s+4}=\dfrac{n+4}{n+4}=1\), which is not the same as the given series.

However, you could restrict the domain of the index such that the series works out. If \(n\) starts at -1 and \(s\) starts at 0, then of course
\[\sum_{n=-1,s=0}^\infty\frac{n+4}{s+4}=\frac{3}{4}+\frac{7}{8}+\cdots\]
but that notation doesn't seem right... In any case, this statement is false.

To check the third statement, use the \(n\)-th term test for divergence.

Answer 2

Thanks :)

Answer 3

You're welcome!