Question

8.
xlogx+x2 log(2x) + x log(3x) +...+x" log(nx) +...the given series is convergent or divergent​

Answer 1

$let un=x(n+1)(log(n+1))p. the u(n+1)=x(n+2)(log(n+2))p now, nlogunun+1=n\log(1x) + np[\log(\log(n(1+1n))-(\log(n(1+2n)))].$

• After that I have used the expansion for log(1+x) and got stuck.
• It seems if I take the limit of

$nlogunun+1, i.e limn→∞nlogunun+1 =∞ (>1)⇒$

• The series seems to be convergent, which is not the answer. Please clarify the mistake and guide me

Answer 2

Answer:xlogx+x2 log(2x) + x log(3x) +...+x" log(nx) +...the given series is convergent .

Step-by-step explanation: xlogx+x2 log(2x) + x log(3x) +...+x" log(nx) +...the given series is convergent.

Convergent series : Convergent series is a series whose partial sums tend to a specific number which is also known as limit . In other terms if the sequence of partial sums is a convergent sequence which means its limit exists and is finite then the series is called convergent series.

Divergent series ; A Divergent series is a series whose partial sums are contrast and do not approach a limit .This series typically goes to -∞ to +∞.In other terms  if the sequence of partial sums is a divergent sequence which means its limit does not exists and is plus or minus infinity then the series is called divergent series.

To know more about limit from the given link

https://brainly.in/question/43105321

To know more about series from the given link

https://brainly.in/question/31539034

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