# Which of the following series are geometric? Which are power ones?

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

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sweererlirumeX

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