Understanding summation decreasing index

Question

I'me following some summation examples and I came to this situation $$|4-4| + \sum_{n=1}^{\infty} |4\cdot0.1^n| = -4+4\sum_{n=0}^{\infty} 0.1^n$$

How do they get to the last result? I thought that $|-4+4|=0$ and decreasing the index should become $\sum_{n=0}^{\infty} |4\cdot0.1^{n+1}|$

What am I doing wrong?

The $\lvert 4-4\rvert$ is a red herring. Ignore it. You add the term for $n = 0$ to the series, therefore you have to subtract it outside. — Daniel Fischer, Jun 27 '13 at 18:43
You can remove all the absolute values, since all the values are non-negative. — Thomas Andrews, Jun 27 '13 at 18:46

score 1 · Accepted Answer · answered Jun 27 '13 at 18:45

1

$|4-4| = -4 + 4 = -4 + 4\cdot (0.1)^0$.

So $$|4-4|+\sum_{n=1}^{\infty} 4\cdot |0.1|^n = -4 + 4\cdot (0.1)^0 + 4\sum_{n=1}^\infty (0.1)^n = -4+4\sum_{n=0}^\infty (0.1)^n$$

answered Jun 27 '13 at 18:45

Thomas Andrews

Thanks Thomas. I now understand. Now just have to figure out that final result is equal to $-4+\frac{4}{1-0.1}$ – Favolas Jun 27 '13 at 18:52
That's just the general equation: $\sum_{n=0}^\infty x^n = \frac{1}{1-x}$ when $|x|<1$. – Thomas Andrews Jun 27 '13 at 18:54
sweet. Many thanks :) – Favolas Jun 27 '13 at 18:55

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