New Notes Blog: `8+6+9/2+27/8+...` Find the sum of the convergent series.

Saturday, 22 February 2014

$\displaystyle{8}+{6}+\frac{{9}}{{2}}+\frac{{27}}{{8}}+\ldots$ Find the sum of the convergent series.

$\displaystyle{8}+{6}+\frac{{9}}{{2}}+\frac{{27}}{{8}}+\ldots\ldots..$

Let's find the common ratio of the terms:

$\displaystyle{r}=\frac{{a}_{{2}}}{{a}_{{1}}}=\frac{{6}}{{8}}=\frac{{3}}{{4}}$

$\displaystyle{r}=\frac{{a}_{{3}}}{{a}_{{2}}}=\frac{{\frac{{9}}{{2}}}}{{6}}=\frac{{9}}{{12}}=\frac{{3}}{{4}}$

So this is a geometric sequence with common ratio of $\displaystyle\frac{{3}}{{4}}$

$\displaystyle{S}_{\infty}=\frac{{a}}{{{1}-{r}}}$ where a is the first term

$\displaystyle{S}=\frac{{8}}{{{1}-\frac{{3}}{{4}}}}$

$\displaystyle{S}=\frac{{8}}{{\frac{{1}}{{4}}}}$

$\displaystyle{S}={32}$

The sum of the given convergent series is 32.

$\displaystyle{8}+{6}+\frac{{9}}{{2}}+\frac{{27}}{{8}}+\ldots\ldots..$

Let's find the common ratio of the terms:

$\displaystyle{r}=\frac{{a}_{{2}}}{{a}_{{1}}}=\frac{{6}}{{8}}=\frac{{3}}{{4}}$

$\displaystyle{r}=\frac{{a}_{{3}}}{{a}_{{2}}}=\frac{{\frac{{9}}{{2}}}}{{6}}=\frac{{9}}{{12}}=\frac{{3}}{{4}}$

So this is a geometric sequence with common ratio of $\displaystyle\frac{{3}}{{4}}$

$\displaystyle{S}_{\infty}=\frac{{a}}{{{1}-{r}}}$ where a is the first term

$\displaystyle{S}=\frac{{8}}{{{1}-\frac{{3}}{{4}}}}$

$\displaystyle{S}=\frac{{8}}{{\frac{{1}}{{4}}}}$

$\displaystyle{S}={32}$

The sum of the given convergent series is 32.

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