Question

help with series

Answer 1

\[\sum_{0}^{\infty} C _{n} X ^{n}\]

Cn+4=Cn for all n>=0
find interval of convergence and formula for F(x)

Answer 2

\[C _{n+4}=C _{n}\]

Answer 3

i am afraid this does not make much sense
saying \(C_n=C_{n+4}\) just tells you every fourth coefficient is the same
not nearly enough information

Answer 4

Nothing more give.
I thing this question should be solved in terms of C.

Answer 5

there must be something missing
for example \(C_k=10 \) for all \(k\) would fit that bill, so would \(C_1=1, C_2=2, C_3=3, C_4=4, C_5-1, C_6=2, ...\)

Answer 6

No nothing at all i check it.

Answer 7

\[F(x) = 1/(1-x ^{4}) (C _{0} + C _{1}x+C _{2}x ^{2}+C _{3}x ^{3})\]
Which converges for x≠1,x≠−1,x≠i,x≠−i

Answer 8

The attack goes like this ...

Answer 9

\[C _{0}+C _{1}x+C _{2}x ^{2}+C _{3}x ^{3}+C _{0}x ^{4}+ C _{1}x ^{5}+C _{2}x ^{6}+C _{3}x ^{7}+C _{0}x ^{8}+C _{1}x ^{9}+C _{2}x ^{10}+C _{3}x ^{11}\]
\[=C _{0}\sum_{n=0}^{\infty}x ^{4n}+C _{1}x \sum_{n=0}^{\infty}x ^{4n}+C _{2}x ^{2} \sum_{n=0}^{\infty}x ^{4n}+C _{3}x ^{3} \sum_{n=0}^{\infty}x ^{4n}+\]
\[= (C _{0}+C _{1}x+C _{2}x ^{2}+C _{3}x ^{3}) \sum_{n=0}^{\infty}x ^{4n}\]
\[[= (C _{0}+C _{1}x+C _{2}x ^{2}+C _{3}x ^{3}) (1/(1-x ^{4})\]
That's it!

Answer 10

Correction: |x| < 1 is the region of convergence.