# series convergence or divergence

#### WartonMorton

Does this series diverge or converge:

1 + [1*2]/[1*3] + [1*2*3]/[1*3*5] + [1*2*3*4]/[1*3*5*7} + ...

#### Failure

Does this series diverge or converge:

1 + [1*2]/[1*3] + [1*2*3]/[1*3*5] + [1*2*3*4]/[1*3*5*7} + ...
This is $$\displaystyle \sum_{n=1}^\infty a_n$$, where $$\displaystyle a_1=1$$, and $$\displaystyle a_n=a_{n-1}\cdot\tfrac{n}{2n-1}$$, and, yes, this series converges.
For consider that $$\displaystyle \lim_{n\to\infty}\frac{n}{2n-1}=\frac{1}{2}<1$$

WartonMorton

#### CaptainBlack

MHF Hall of Fame
Does this series diverge or converge:

1 + [1*2]/[1*3] + [1*2*3]/[1*3*5] + [1*2*3*4]/[1*3*5*7} + ...
$$\displaystyle a_{n+1}=a_n \frac{n+1}{2n+1}$$

so:

$$\displaystyle \frac{a_{n+1}}{a_n}=\frac{1}{2-\frac{1}{n+1}}$$

CB

WartonMorton

#### Prove It

MHF Helper
Does this series diverge or converge:

1 + [1*2]/[1*3] + [1*2*3]/[1*3*5] + [1*2*3*4]/[1*3*5*7} + ...
$$\displaystyle 1 + \frac{1 \cdot 2}{1 \cdot 3} + \frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5} + \frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7} + \dots$$

$$\displaystyle = \sum_{k = 1}^{\infty}{\frac{k!}{\frac{(2k)!}{k!2^k}}}$$

$$\displaystyle = \sum_{k = 1}^{\infty}\frac{(k!)^22^k}{(2k)!}$$

Using the ratio test:

$$\displaystyle \lim_{n \to \infty}\left|\frac{t_{n + 1}}{t_n}\right| = \lim_{n \to \infty}\left|\frac{\frac{[(n + 1)!]^22^{n + 1}}{[2(n + 1)]!}}{\frac{(n!)^22^n}{(2n)!}}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left|\frac{(2n)![(n + 1)!]^22^{n + 1}}{[2(n + 1)]!(n!)^22^n}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left|\frac{2(2n)!(n + 1)^2(n!)^22^n}{(2n + 2)(2n + 1)(2n)!(n!)^22^n}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left|\frac{2(n + 1)^2}{2(n + 1)(2n + 1)}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left|\frac{n + 1}{2n + 1}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left(\frac{n + 1}{2n + 1}\right)$$

$$\displaystyle = \lim_{n \to \infty}\frac{1}{2}$$ by L'Hospital's Rule

$$\displaystyle = \frac{1}{2}$$

$$\displaystyle < 1$$.

So the series is convergent.

WartonMorton

#### CaptainBlack

MHF Hall of Fame
$$\displaystyle 1 + \frac{1 \cdot 2}{1 \cdot 3} + \frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5} + \frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7} + \dots$$

$$\displaystyle = \sum_{k = 1}^{\infty}{\frac{k!}{\frac{(2k)!}{k!2^k}}}$$

$$\displaystyle = \sum_{k = 1}^{\infty}\frac{(k!)^22^k}{(2k)!}$$

Using the ratio test:

$$\displaystyle \lim_{n \to \infty}\left|\frac{t_{n + 1}}{t_n}\right| = \lim_{n \to \infty}\left|\frac{\frac{[(n + 1)!]^22^{n + 1}}{[2(n + 1)]!}}{\frac{(n!)^22^n}{(2n)!}}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left|\frac{(2n)![(n + 1)!]^22^{n + 1}}{[2(n + 1)]!(n!)^22^n}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left|\frac{2(2n)!(n + 1)^2(n!)^22^n}{(2n + 2)(2n + 1)(2n)!(n!)^22^n}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left|\frac{2(n + 1)^2}{2(n + 1)(2n + 1)}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left|\frac{n + 1}{2n + 1}\right|$$

$$\displaystyle = \lim_{n \to \infty}\left(\frac{n + 1}{2n + 1}\right)$$

$$\displaystyle = \lim_{n \to \infty}\frac{1}{2}$$ by L'Hospital's Rule

$$\displaystyle = \frac{1}{2}$$

$$\displaystyle < 1$$.

So the series is convergent.
This looks like very hard work compared to the two previous posts, I expect that explains why it took 11 minutes longer to write.

CB

WartonMorton