# Integration

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Please explain how to solve the attached problems (as much explanation as possible) and solve to the specified answers.

Find a formula for the sum of n terms. Use the formula to find the limit as n approaches infinity.

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1. Find

First we need find

Then we have to apply the limit.

Summation of I=1 to n = 16 [ 1/n^2+ 2/n^2+......n/n^2]

= 16/n^2 [ 1+2+3+..........+n]

= 16/ n^2 [Sn] -------------- ------ A

Let us take Sn = 1+2+3+........n

We know that Sn, 1+2+3+.......+n = {n(n+1)}/2

We can prove this by mathematical induction method.

LHS = 1+2+.....+n

RHS = {n(n+1)}/2

Put n = 1

LHS =1 and RHS = {1(1+1)}/2 => 2/2 =1

Therefore, LHS = RHS

S(1) is true.

Hence, S(k) is true.

(ie) 1+2+3+.....+k = {k(k+1)}/2

We shall now ...

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