Solve Sum Series Questions: Find Interval of Convergence

[sweetsurf3]

infin
1.) Suppose that (6x)/(10+x)= sum C_n(X^(n))
n=o to

Find: (what I got, but was wrong)
C_0= 0
C_1= 6/11
C_2= 1
C_3= 18/13
C_4= 24/14
Find: Radius of convergence R, I got 10.

I've been struggling with setting any sum problem up.



2) Find the interval of convergence for the given power series: n=1
(x-5)^(n)/n(-10)^(n)
The series is convergent from x=___, left end included (Y,N)___
to x=___, right end included (Y,N)___


any info and pointers is GREATLY appreciated, thank you :)

 

[HallsofIvy]

infin
1.) Suppose that (6x)/(10+x)= sum C_n(X^(n))
n=o to

Find: (what I got, but was wrong)
C_0= 0
C_1= 6/11
C_2= 1
C_3= 18/13
C_4= 24/14
Find: Radius of convergence R, I got 10.

I've been struggling with setting any sum problem up.



2) Find the interval of convergence for the given power series: n=1
(x-5)^(n)/n(-10)^(n)
The series is convergent from x=___, left end included (Y,N)___
to x=___, right end included (Y,N)___


any info and pointers is GREATLY appreciated, thank you :)

I would notice that 6/(10+ x)= 0.6(1/(1- x/10)) and recognise that as the sum of a geometric sequence: $\Sum_{i= 0}^{\infty} 0.6(x/10)^i$- which converges as long as |x/10|< 1 or |x|< 10.

Of course, your sum is 6x/(1+ x). Okay, multiply the geometric sum by x: $\Sum{i=0}^{\infty}0.6 x^{i+1}/10^i$. That won't affect the radius of convergence.

To determine whether the series converges at each of the end points, you might want to see whether the series is "alternating" or not.