Question

Find the open interval on which the given power series converges absolutely.

E (2^n-2^(-n))(x-1)^n

Answer 1

take the limit of an/an-1 as n approaches infinity

Answer 2

\[\lim_{n\to\ inf}\frac{ 2^n(1-2^{-2n})(x-1)^n} {2^{n-1} (1-2^{-2(n-1)})(x-1)^{n-1}}\]
\[\lim_{n\to\ inf}\frac{ 2(1-2^{-2n})(x-1)} {1-2^{-2n+1}}\]
\[2|x-1|\ \lim_{n\to\ inf}\frac{1-2^{-2n}} {1-2^{-2n+1}}\]

Answer 3

looks to me like the n parts limit off to 1/1

sooo, if i did this right

2|x-1| < 1

|x-1| < 1/2

-1/2 < x-1 < 1/2

1/2 < x < 3/2