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Identify whether these series are divergent or convergent geometric series and find the sum, if possible.

Identify Whether These Series Are Divergent Or Convergent Geometric Series And Find The Sum If Possible class=

Sagot :

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A geometric series:
[tex]\sum^{\infty}_{i=1}=a_1 \times r^{i-1}[/tex]
It's convergent if |r|<1.
It's divergent if |r|≥1.
The sum can be found if it's a convergent series; it's equal to [tex]\frac{a_1}{1-r}[/tex].

3.
[tex]\sum^{\infty}_{i=1} 12 (\frac{3}{5})^{i-1} \\ \\ a_1=12 \\ r=\frac{3}{5} \\ \\ |r|<1 \hbox{ so it's convergent} \\ \\ \sum^{\infty}_{i=1} 12 (\frac{3}{5})^{i-1}=\frac{12}{1-\frac{3}{5}}=\frac{12}{\frac{5}{5}-\frac{3}{5}}=\frac{12}{\frac{2}{5}}=12 \times \frac{5}{2}=6 \times 5=30[/tex]

The answer is: This is a convergent geometric series. The sum is 30.

4.
[tex]\sum^{\infty}_{i=1} 15(4)^{i-1} \\ \\ a_1=15 \\ r=4 \\ \\ |r| \geq 1 \hbox{ so it's divergent}[/tex]

The answer is: This is a divergent geometric series. The sum cannot be found.