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grin007 [14]
3 years ago
10

Please help me solve this​

Mathematics
1 answer:
Viktor [21]3 years ago
3 0

You can factor the 32 out of the sum:

\displaystyle \sum_{m=1}^\infty 32 \cdot \left(\dfrac{1}{2}\right)^{m-1} = 32\sum_{m=1}^\infty \left(\dfrac{1}{2}\right)^{m-1}

We can also change the index as follows

\displaystyle 32\sum_{m=1}^\infty \left(\dfrac{1}{2}\right)^{m-1} = 32\sum_{m=0}^\infty \left(\dfrac{1}{2}\right)^{m}

Now, we have a theorem that states that the series

\displaystyle \sum_{m=1}^\infty a^m

converges if and only if |a|, and in this case we have

\displaystyle \sum_{m=1}^\infty a^m = \dfrac{1}{1-a}

This is your case, because you have

|a|=\dfrac{1}{2}

which implies that your series converges, and the value is

\displaystyle 32\sum_{m=0}^\infty \left(\dfrac{1}{2}\right)^{m} = 32 \cdot \dfrac{1}{1-\frac{1}{2}} = 32\cdot 2 = 64

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