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Consider the following geometric series:

$\displaystyle {\left(-{x}-{4}\right)}$ $\displaystyle +$ $\displaystyle {\left({4}{x}^{{2}}+{15}{x}-{4}\right)}$ $\displaystyle +$ $\displaystyle {\left(-{16}{x}^{{3}}-{56}{x}^{{2}}+{31}{x}-{4}\right)}\ldots$

a) Determine the common ratio of the sequence in terms of $\displaystyle {x}$ .

$\displaystyle {r}=$

$\displaystyle =$

(2)

b) For which value of $\displaystyle {x}$ will the series converge?

Convergence Condition in terms of $\displaystyle {r}$ :

(substitute $\displaystyle {r}$ )

$\displaystyle {x}\in$

(3)

c) Calculate $\displaystyle {S}_{{\infty}}$ if $\displaystyle {x}=\frac{{1}}{{5}}$

Formula

$\displaystyle {S}_{\infty}=$

Substitution

$\displaystyle {S}_{\infty}=$

Solve

$\displaystyle {S}_{\infty}=$

(3)

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Consider the following geometric series:

a) Determine the common ratio of the sequence in terms of x\displaystyle {x}x .

b) For which value of x\displaystyle {x}x will the series converge?

c) Calculate S∞\displaystyle {S}_{{\infty}}S∞​ if x=15\displaystyle {x}=\frac{{1}}{{5}}x=51​

a) Determine the common ratio of the sequence in terms of $\displaystyle {x}$ .

b) For which value of $\displaystyle {x}$ will the series converge?

c) Calculate $\displaystyle {S}_{{\infty}}$ if $\displaystyle {x}=\frac{{1}}{{5}}$