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

hexacordoK 2021-08-16 Answered
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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Expert Answer

sweererlirumeX
Answered 2021-08-17 Author has 12380 answers

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.

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