Find the values of x for which
$$\sum_{n = 1}^{\infty} \frac{2^n}{x^n}$$ converges.
This is what I'm thinking:
I tried graphing it with a really big n number to get an idea on how it may look, and the graph shows nothing. Im completely stuck, am I not considering a theorem?
$\endgroup$ 22 Answers
$\begingroup$The series
$$\sum_{n = 1}^{\infty} \frac{2^n}{x^n} = \sum_{n = 1}^{\infty} \left(\frac{2}{x}\right)^n$$
is a geometric series with initial term $2/x$ and common ratio $2/x$.
A geometric series with a non-zero initial term converges when the common ratio has absolute value less than $1$.
You could also apply the Ratio Test, which leads to the same result, although you have to check that the series diverges when $x = \pm 2$.
$\endgroup$ $\begingroup$Hint 1 : Look at $$\sum_{n=1}^\infty (\frac{2}{x}) ^n$$
which is the same sum.
Hint 2 : The given series is a geometric series.
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