I tried graphing it with a really huge n number to get an idea on how it may look, and the graph mirrors nothing. Im totally stuck, to be I no considering a theorem?


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The collection

$$sum_n = 1^infty frac2^nx^n = sum_n = 1^infty left(frac2x ight)^n$$

is a geometric collection with initial term $2/x$ and also common ratio $2/x$.

A geometric collection with a non-zero initial hatchet converges as soon as the common ratio has absolute value less than $1$.

You could likewise apply the ratio Test, which leads to the very same result, return you have actually to check that the collection diverges once $x = pm 2$.

You are watching: Find all values of x for which the series converges


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Hint 1 : Look at $$sum_n=1^infty (frac2x) ^n$$

which is the exact same sum.

Hint 2 : The given collection is a geometric series.


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