1
$\begingroup$

Are these expressions equivalent?

$$ \sum_{n=1}^\infty nc_n(x-1)^n \qquad\text{and}\qquad \sum_{n=0}^\infty nc_n(x-1)^n $$

My reasoning is that they both start at zero the first goes the following pattern:

Pattern 1 $$\sum_n^\infty a_n= c_1(x-1)^1+2c_2(x-1)^2\dots$$

Pattern 2 $$\sum_n^\infty b_n= 0+ c_1(x-1)^1+2c_2(x-1)^2\dots$$

Please let me know if my logic is sound please? If they aren't could you please explain?

I added this differential equation tag because it belongs to a DE I am manipulating the indexes for.

$\endgroup$
  • 4
    $\begingroup$ Your logic is correct $\endgroup$ – Ninad Munshi May 15 at 14:42
1
$\begingroup$

Yes, you are correct.

Just be cautious that $c_n$ should be a term that you expect that it can be evaluated with value $n=0$, say $c_n=\frac1n$ shouldn't be there.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.