Sunday 20 September 2015

summation - An equality: from one sum to 2 sums.

I have an equality:
$$
\ddot{a}_{30} = \frac{1}{0.75}\left( \sum_{k=0}^{\infty} \left(\frac{1}{1.06}\right)^k \left( 1 - \frac{30+k}{120}\right) \right)=\left( \sum_{k=0}^{\infty} \left(\frac{1}{1.06}\right)^k - \frac{1}{90} \sum_{k=0}^{\infty} k\left(\frac{1}{1.06}\right)^k \right)
$$



How from $\frac{1}{0.75}\left( \sum_{k=0}^{\infty} \left(\frac{1}{1.06}\right)^k \left( 1 - \frac{30+k}{120}\right) \right)$ we get $\left( \sum_{k=0}^{\infty} \left(\frac{1}{1.06}\right)^k - \frac{1}{90} \sum_{k=0}^{\infty} k\left(\frac{1}{1.06}\right)^k \right) ?$



Because I do not understand where we lost $\frac{1}{0,75}$ in the first sum.

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