# Thread: Using the binomial expansion.

1. ## Using the binomial expansion.

Any faulty logic?

Prove: if a>1, (a^n)/n --> infinity.

Since a>1 we can write:

a = [(1 + k)]/ [n^(1/n)];

n*a^n = (1 + k)^n;

n*a^n =1 + nk + [n(n -1 )(k^2)]/2!+.....;

So, a^n =1/n + k +.....;

>1/n + k;

>M, for M>0 for say 1/n > M.

Is there a problem here since n < 1/M? Or, does this still imply n > M, so it doesn't matter?

Thanks.

2. Originally Posted by cgiulz
Any faulty logic?

Prove: if a>1, (a^n)/n --> infinity.

Since a>1 we can write:

a = [(1 + k)]/ [n^(1/n)];

n*a^n = (1 + k)^n;

n*a^n =1 + nk + [n(n -1 )(k^2)]/2!+.....;

So, a^n =1/n + k +.....;
At this step you should have,
$\frac{a^n}{n} = \frac{1}{n} + k + \frac{1}{2}k^2(n-1) + ... \geq \frac{1}{2}k^2(n-1)$

Now certainly, $\frac{1}{2}k^2(n-1)\to \infty$.
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