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Test the following series for convergence $\sum_{n=1}^{\infty} 3\left(\frac{1}{2}\right)^{n-1}$

$\sum_{n=1}^{\infty} a r^{n-1}$

Gemetric series (G.s) with $r=\frac{1}{2}<1$

convergent series

$a=3 \cdot\left(\frac{1}{2}\right)^{1-1}=3\left(\frac{1}{2}\right)^{0}=3(1)=3$

sum $=\frac{a}{1-r}=\frac{3}{1-\frac{1}{2}}=6$

Is the following series convergent? $\sum_{n=1}^{\infty} 3\left(\frac{1}{2}\right)^{n}$

Geometric series with $r=\frac{1}{2}<1$

Convergent series

$a=3\left(\frac{1}{2}\right)^{1}=\frac{3}{2}$

Sum $=\frac{a}{1-r}=\frac{\frac{3}{2}}{1-\frac{1}{2}}=\frac{\frac{3}{2}}{\frac{1}{2}}=3$

Is the following series convergent? $\sum_{n=2}^{\infty} 3\left(\frac{1}{2}\right)^{n+1}$

Gometric series (G.S) with $r=\frac{1}{2}<1$

Convergent series

$a=3\left(\frac{1}{2}\right)^{2+1}=3\left(\frac{1}{2}\right)^{3}=3\left(\frac{1}{8}\right)=\frac{3}{8}$

sum $=\frac{a}{1-r}=\frac{\frac{3}{8}}{1-\frac{1}{2}}=\frac{\frac{3}{8}}{\frac{1}{2}}=\frac{3}{8} \times \frac{2}{1}=\frac{3}{4}$

Is the following series convergent? $\sum_{n=0}^{\infty}(-\ln (2))^{n}$

Gometric series with $|r|=|-\ln (2)|=\ln (2)=0.69 \ldots<1$

convergent series

$a=(-\ln (2))^{0}=1$

sum $=\frac{a}{1-r}=\frac{1}{1-(-\ln (2))}=\frac{1}{1+\ln (2)}$

Is the following series convergent? $\sum_{n=0}^{\infty}\left[\cos \left(\frac{\pi}{3}\right)\right]^{n}$

Geometric series with $|r|=\left|\cos \frac{\pi}{3}\right|=\frac{1}{2}<1$

Convergent series

$a=\left[\cos \frac{\pi}{3}\right]^{0}=1$

sum $=\frac{a}{1-r}=\frac{1}{1-\frac{1}{2}}=\frac{1}{\frac{1}{2}}=2$

$\sum_{n=0}^{\infty}\left[\cos \frac{\pi}{3}\right]^{n}=\sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^{n}$

Find the value of t such that $\sum_{n=1}^{\infty}\left(\frac{t}{t+2}\right)^{n}=3$

Geometric series (G.S) $=\sum_{n=1}^{\infty}\left(\frac{t}{t+2}\right)^{n}$ with $r=\frac{t}{t+2}$

$a=\left(\frac{t}{t+2}\right)^{1}=\frac{t}{t+2}$

Sum $=\frac{a}{1-r}=\frac{\frac{t}{t+2}}{1-\frac{t}{t+2}}$

$=\frac{\frac{t}{t+2}}{\frac{t+2}{t+2}-\frac{t}{t+2}}=\frac{\frac{t}{t+2}}{\frac{t+2-t}{t+2}}$

Sum $=\frac{t}{2}=3 \quad \Rightarrow t=3 \cdot 2=6$

$t=6$