Answer to Question #264663 in Real Analysis for Dhruv bartwal

Question

Evaluate

Lim 3nΣr=1 n^2/(4n+r)^3

n→∞

Answer 1

Let the solution of our equation be of the form y=\displaystyle\sum_{n=0}^{\infty}a_nx^ny=n=0∑∞​an​xn .....(1)

From which,

y'=\displaystyle\sum_{n=0}^{\infty}na_nx^{n-1}=\displaystyle\sum_{n=1}^{\infty}na_nx^{n-1}y′=n=0∑∞​nan​xn−1=n=1∑∞​nan​xn−1

y''=\displaystyle\sum_{n=0}^{\infty}n(n-1)a_nx^{n-2}=\displaystyle\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}y′′=n=0∑∞​n(n−1)an​xn−2=n=2∑∞​n(n−1)an​xn−2

We then substitute the above expressions in our equation

\displaystyle\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}-\displaystyle\sum_{n=1}^{\infty}na_nx^{n}+\displaystyle\sum_{n=0}^{\infty}a_nx^n=0n=2∑∞​n(n−1)an​xn−2−n=1∑∞​nan​xn+n=0∑∞​an​xn=0

\displaystyle\sum_{n=0}^{\infty}(n+1)(n+2)a_{n+2}x^{n}-\displaystyle\sum_{n=1}^{\infty}na_nx^n+\displaystyle\sum_{n=1}^{\infty}a_nx^n+a_0=0n=0∑∞​(n+1)(n+2)an+2​xn−n=1∑∞​nan​xn+n=1∑∞​an​xn+a0​=0

2a_2+6a_3x+\displaystyle\sum_{n=1}^{\infty}[(n+1)(n+2)a_{n+2}-na_n+a_n]x^n+a_0=02a2​+6a3​x+n=1∑∞​[(n+1)(n+2)an+2​−nan​+an​]xn+a0​=0

Comparing both sides of equal powers of x we have,

2a_2+a_0=0\implies a_2=-\frac{1}{2}a_02a2​+a0​=0⟹a2​=−21​a0​

6a_3=0\implies a_3=06a3​=0⟹a3​=0