hexacordoK

## Answered question

2021-08-16

Which of the following series are geometric? Which are power ones?
a) $1+\frac{x}{2}+\frac{{x}^{2}}{4}+\frac{{x}^{3}}{8}+\frac{{x}^{4}}{16}+\dots$
b) $1+1.1+1.21+1.331+1.4641+1.6105+\dots$
c) ${\left(\frac{1}{3}\right)}^{2}+{\left(\frac{1}{3}\right)}^{4}+{\left(\frac{1}{3}\right)}^{6}+{\left(\frac{1}{3}\right)}^{8}+\dots$
d) $1+x+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}+\dots$
e) $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\dots$
f) $\frac{1}{{x}^{2}}+\frac{1}{x}+1+x+{x}^{2}+{x}^{3}+{x}^{4}+\dots$

### Answer & Explanation

sweererlirumeX

Skilled2021-08-17Added 91 answers

a) $1+\frac{x}{2}+\frac{{x}^{2}}{4}+\frac{{x}^{3}}{8}+\frac{{x}^{4}}{16}+\dots$ is both power and geometric series.
b) $1+1.1+1.21+1.331+1.4641+1.6105+...$ is only power series
c) ${\left(\frac{1}{3}\right)}^{2}+{\left(\frac{1}{3}\right)}^{4}+{\left(\frac{1}{3}\right)}^{6}+{\left(\frac{1}{3}\right)}^{8}+\dots$ is geometric series.
d) $1+x+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}+\dots$ is geometric series.
e) $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\dots$ is neither geometric nor power one.
f) $\frac{1}{{x}^{2}}+\frac{1}{x}+1+x+{x}^{2}+{x}^{3}+{x}^{4}+\dots$ is power series.

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