Question

Quick question about series...

How would you go from:

∑k=0inf2(k+2)(k+1)ck+2xk+∑k=1infkckxk+∑k=0infckxk


to

\[\4c _{2} + c _{0} + sum_{k=1}^{inf}[2(k+2)(k+1)c _{k+2}+kc _{k}+c _{k}]x ^{k}\]

I know we factored out the constant but 4c2 + c0 doesn't make too much sense to me. any help?

Answer 1

∑(k=0..inf) [2(k+2)(k+1)c_k +2x^k]
+ ∑(k=1..inf) [k c_k x^k]
+ ∑(k=0..inf) [c_k x^k]

\[\sum_{0}[2(k+2)(k+1)c_k+2x^k]+\sum_{1}[k~c_kx^k]+\sum_{0}[c_kx^k]\]
is this what you got?

Answer 2

if we want to get the index under the summations lined up, we need to pull out the 0th term so that they all have a starting index of 1

\[\sum_{0}[2(k+2)(k+1)c_k+2x^k]\\
~~~~~~~~~~~~~~=2(0+2)(0+1)c_0+2x^0+\sum_{1}[2(k+2)(k+1)c_k+2x^k]\\
~~~~~~~~~~~~~~=2(2)(1)c_0+2+\sum_{1}[2(k+2)(k+1)c_k+2x^k]\]
\[\sum_{0}[c_kx^k]\\
~~~~~~~~~~~~~~=c_0x^0+\sum_{1}[c_kx^k]\\
~~~~~~~~~~~~~~=c_0+\sum_{1}[c_kx^k]\\
\]

but its hard to read what you posted but this is the general process that i think they applied.