Consider the series?

Question

Consider the series:
the sum from n=1 to infinity of ((10^n)(x^n)(n+1))/(n+5)

for the ratio test i got ((10x)(n+2)(n+5))/(n+6)
i am confused on what to do next.

Answer 1

AGAIN, let n=>oo
we get infinity for your expression.
so we get L>1 fro all x
therefore DIVERGES for all x

But, I get a dif't expression
I get I [10x (n+2)(n+5)] / [ (n+1)(n+6) ] I
which goes to I10xI as n=>oo
we need to keep I10xI < 1 in order to converge
-1/10 < x < 1/10

test endpoints if you need interval
radius is 1/10

:)

Answer 2

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Answer 3

[(10^n)(x^n)(n+1)] / (n+5)

= [(10x)^n] * [(n+1)/(n+5)]

The right hand term (n+1)/(n+5) →1 as n→∞. So whether it converges or diverges depends on the left hand term.

If |10x| < 1 it will converge. Otherwise it will diverge. So the series converges for x such that |x| < 1/10.