Question

classify each series as convergent or divergent. divergent convergent 1 + 2+4 + 8+... 200 + 100+50 + 25+... 1/2 + 2/3+8/9 + 32/27+... 10 + 5+0 + - 5+... 4 - 2+1 - 1/2+... 4 - 8+16 - 32+...

Explanation:

Step1: Recall geometric - series formula

A geometric series is of the form $\sum_{n = 0}^{\infty}a\cdot r^{n}$, and it converges if $|r|\lt1$ and diverges if $|r|\geq1$.

Step2: Analyze $1 + 2+4 + 8+\cdots$

Here, $a = 1$ and $r = 2$. Since $|r|=2\gt1$, the series diverges.

Step3: Analyze $200 + 100+50 + 25+\cdots$

Here, $a = 200$ and $r=\frac{1}{2}$. Since $|r|=\frac{1}{2}\lt1$, the series converges.

Step4: Analyze $\frac{1}{2}+\frac{2}{3}+\frac{8}{9}+\frac{32}{27}+\cdots$

The ratio between consecutive terms is $\frac{\frac{2}{3}}{\frac{1}{2}}=\frac{4}{3}$. Since $|r|=\frac{4}{3}\gt1$, the series diverges.

Step5: Analyze $10 + 5+0 - 5+\cdots$

The common difference between consecutive terms is not a constant ratio. But we can rewrite it as a geometric - like series. The ratio between the first two terms is $\frac{5}{10}=\frac{1}{2}$, between the second and third is $0$, so it is not a geometric series. However, we can observe that the terms do not approach a finite value as $n$ increases, so it diverges.

Step6: Analyze $4-2 + 1-\frac{1}{2}+\cdots$

Here, $a = 4$ and $r=-\frac{1}{2}$. Since $|r|=\frac{1}{2}\lt1$, the series converges.

Step7: Analyze $4-8 + 16-32+\cdots$

Here, $a = 4$ and $r=-2$. Since $|r| = 2\gt1$, the series diverges.

Answer:

a. Divergent
b. Convergent
c. Divergent
d. Divergent
e. Convergent
f. Divergent