Question

Hi everyone! Need alot of help on this one...
f(x) has the Power Series 5+10x+15x^2+20x^3+25x^4+... where -infinity 13 years ago

Answer 1

Whoever tackles this one, could you please help me with the steps and the intuition behind them? Thanks!

Answer 2

I am going to write down the steps and answer, but I need help understanding them...one second...

Answer 3

ok

Answer 4

\[\int\limits_{0}^{x} 5+10w+15w^2+20w^3 dw\]

then it goes...

Answer 5

wait...does the series end? or is it infinite?

Answer 6

\[5w+5w^2+5w^3 | from 0 \to x\]

Answer 7

\[5x+5x^2+5x^3+5x^4\] this is the final answer

Answer 8

none of this makes any sense to me! lol

Answer 9

oh it also gave a hint that said assume the constant of integration is 2...which doesn't help at all

Answer 10

here's what I think is going on

f(x) = 5+10x+15x^2+20x^3+25x^4+... (infinite power series)

f(w) = 5+10w+15w^2+20w^3+25w^4+... (infinite power series)

f(w) = (5+5n)w^n ... where n ranges from n = 0 to n = infinity

int(f(w)dw) = int( (5+5n)w^n )

int(f(w)dw) = (5+5n)/(n+1)w^(n+1) + C

int(f(w)dw) = (5(1+n))/(n+1)w^(n+1) + C

int(f(w)dw) = 5w^(n+1) + C .. where n goes from n = 0 to n = infinity

Answer 11

Now evaluate at the endpoints from w = 0 to w = x to get

[5x^(n+1) + C] - [5(0)^(n+1) + C]

5x^(n+1) + C - 5(0)^(n+1) - C

5x^(n+1)

Answer 12

So...