Question

Sum function..

Answer 1

Can someone help me find the sum function to:
\[\sum_{n=0}^{∞}(a*n+b)x^n\]
:)

Answer 2

depends on what your value of x is, if |x| < 1, then separate the bracket, first term will be arithmetic-geometric series and second will be geometric series

Answer 3

if |x|>=1, then it diverges

Answer 4

I have been given a hint: \[\sum_{n=0}^{∞}x^n=1/(1-x)\]
And
\[\sum_{n=0}^{∞}\frac{ 1 }{ (1-x)^2 }\]

Answer 5

\[\sum( an+b)x^n\]
\[\sum an~x^n+bx^n\]
\[\sum an~x^n+\sum bx^n\]
\[a\sum n~x^n+b\sum x^n\]
not real sure where thats going tho

Answer 6

your given hint applies if and only if |x|<1

Answer 7

I must find the answer for all pairs of constants a and b (not both zero). It shall also indicate for which x formula is valid.

Answer 8

the condition is i stated it above.
\[ \sum_{n=0}^\infty nx^n = \sum_{n=0}^\infty x \frac{d(x^n)}{dx} = x \frac{d(\sum x^n)}{dx} = x \; \frac{d}{dx}(1/(1-x))\]
you can interchange the order of differentiation because for |x|<1, the series converges uniformly.
also there is a quick trick without differentiation.

Answer 9

Thank you @experimentX and @amistre64

Answer 10

yw