Question
Asked Nov 16, 2019
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I am working with power series and this answer explanation shown in the screen shot does not make sense to me. Why and are they reindexing and how to they get that resulting answer?

(- 1)k + 1 1 0k- 1 kx2k-1 = 0 +x -20x3 300x5 -40000x7 + ...
in its expanded
The power series for g(x) is
k 0
form. Drop the zero term and re-index the series so that k 0 corresponds to the term x
0o
g(x)>(-1)k10K (k+ 1)x2k+1
k 0
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(- 1)k + 1 1 0k- 1 kx2k-1 = 0 +x -20x3 300x5 -40000x7 + ... in its expanded The power series for g(x) is k 0 form. Drop the zero term and re-index the series so that k 0 corresponds to the term x 0o g(x)>(-1)k10K (k+ 1)x2k+1 k 0

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Expert Answer

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Step 1

Note that the variable k  lies inside the summation as a product term...

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Σ-)0 ke- (-1)** 10*' (0) >20 Σ-1)" k+1 = (-1)'10-1(0)x20)-1 2k-1 Σ-1)"10*" kκ. k-0 k-1 k+1 -1x10' x 0x x1 +1)10-1 κ 2k-1 k1 k+1 0(-1) 10*-1kx2*-1 k1 k+1 -Σ-1)104" kλ- Ε k-1

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