Question

Show that the following series converges: (4)
(2n 3)(2n 5)
1
n 1 + + ∑
∞
=

Answer 1

Proof:

We need to show that the series converges.

Let $\displaystyle\sum_{i=1}^n$$\frac {1} {(2n+3) (2n+5)} =$ $\displaystyle\sum_{i=1}^n$ u_i Here u_n =$\frac {1} {(2n+3) (2n+5)}$

Let us consider the series $\displaystyle\sum_{i=1}^n$ v_i here v_n =$\frac {1} {n^2}$

Now, $\lim\limits_{n->\infin}$$\frac {} { }$$\frac {u_n} {v_n}$ = $\lim\limits_{n->\infin}$ $\frac {\frac {1} {[(2n+3) (2n+5)]}} {\frac {1} {(n^2)}}$ = $\lim\limits_{n->\infin}$ $\frac {n^2} {(2n+3) (2n+5)}$

= $\lim\limits_{n->\infin}$$\frac {1} {(2+\frac {3} {n} ) (2+\frac {5} {n})}$ = $\frac {1} {(2+0) (2+ 0) }$ ( Here $\frac {3} {\infin } = 0$ and $\frac {5} {\infin } = 0$ )

= $\frac {1} {4} < 1$

$\lim\limits_{n->\infin}$$\frac {u_n} {v_n} < 1$ , then the given series converges.

Answer: the series converges.