Rewrite the given expression as a single Power series

Question

anonymous · Answer

$$\sum_{n=1}^{\infty}2nc_nx^{n-1}+\sum_{n=0}^{\infty}6c_nx^{n+1}$$

anonymous · Answer

$$k=n-1$$

$$k+1=n$$

$$\sum_{k=0}^\infty2(k+1)c_{k+1}x^{k+1-1}$$

anonymous · Answer

$$\sum_{k=0}^\infty2(k+1)c_{k+1}x^k+\sum_{n=0}^\infty6c_nx^{n+1}$$

anonymous · Answer

let k=n+1

$$k-1=n$$

$$\sum_{k=0}^\infty2(k+1)c_{k+1}x^k+\sum_{k=1}^\infty6c_{k-1}x^{k}$$

anonymous · Answer

@UnkleRhaukus

anonymous · Answer

move the first power series up one

$$2(0+1)c_1x^0+\sum_{k=1}^\infty2(k+1)c_{k+1}x^k+\sum_{k=1}^\infty6c_{k-1}x^k$$\]

anonymous · Answer

$$2c_1+\sum_{k=1}^\infty[2(k+1)c_{k+1}+6c_{k-1}]x^k$$

anonymous · Answer

have you done this yet @UnkleRhaukus

unklerhaukus · Answer

i havent done this stuff recently,
 your work looks good

anonymous · Answer

WAit you haven't done " Using power series to solve differential equaions?"

unklerhaukus · Answer

i dont think i have , sounds interesting

unklerhaukus · Answer

so is that your final solution?

anonymous · Answer

yep

anonymous · Answer

i can split it up more but this is just writing them as one... doing the differential equations is much different