How can you compute the limit of
∞∑n=1n(2/3)n
Evidently it is equal to 6 by wolfram alpha but how could you compute such a sum analytically?
Answer
∞∑n=1n(2/3)n=∞∑m=1∞∑n=m(2/3)n=∞∑m=1(2/3)m1−2/3=2/3(1−2/3)2=6.
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