Determine which one of the following tests could be used to show that the series the summation from n equals 1 to infinity of 1 over n squared is convergent.

Question

Determine which one of the following tests could be used to show that the series  is convergent.

A	Geometric series test
B	Integral test
C	Nth term test
D	Ratio test

Answer 1

I will explain for you here that we can use integral test

suppose f is a continuous , positive and decreasing on the interval [ 1, ∞) , and let a_n = f (n) . Then the series

∞

∑ a_n is convergent if and only if the improper integral

n=1

∞

∫ f(x) dx is convergent.

1

Since 1/ (n+1 )² < 1 / n ² the series is decreasing, or you can prove that by taking the first derivative for f(n) and you will find it negative]

Now

∞ t ∞

∫ 1/x² dx = lim ∫ 1/x² dx = lim -1/x ⌉ = - lim [ 1/t - 1] = 1 convergent

1 t→∞ 1 t → ∞ 1 t → ∞

Hence

∞

∑ 1/ n² is convergent by the integral test.

n=1