Question

mple 1 : Prove that Σ 1/n square +1 converges​

Answer 1

Answer:

∑n=0 1(n+1)(n+2)

is convergent, based on the direct comparison test.

Step-by-step explanation:

The series:

∞∑n=0 1(n+1)(n+2)

has positive terms, so we can use the direct comparison test identifying another convergent series:

∞∑n=0 bn

such that:

1(n+1)(n+2)<bn for n>N.

Now note that clearly, for

n>1:

(n+1)>n

⇒1n+1<1n

and also:

(n+2)>n

⇒1n+2<1n

and multiplying the two inequalities:

1(n+1)(n+2)<1n2 for n>

Now,

∞∑n=1

1n2=π26

is convergent, so also our series is convergent

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