The geometric series is as follows :
$$n/2^n$$
I am using the ratio test therefore comparing : $$(n+1/2^{n+1})/ (n/2^n)$$
my next line of work is :
$$(n+1/2^{n+1}) * (2^n/n)$$
however I am not familiar with multiplying fractions of this sort, so cannot continue.
my tutor has the next line as .. $$Un+1/Un = (n+1/n) * (1/2)$$
which I do not understand.
Please could someone explain the steps to me in layman's terms so I can deduce an answer.
Any help is appreciated.
On
Your expression for the ratio (which is correct) is $((n+1)/2^{n+1})\cdot(2^n/n)$, which equals $((n+1)/n)\cdot (2^n/2^{n+1})$. Now, $2^n/2^{n+1}={1\over 2}$, and $(n+1)/n=1+1/n$. Your expression for the ratio therefore simplifies to $(1+1/n)\cdot {1\over 2}$. This ratio is $<1$ for sufficiently large $n$. Indeed, $n\ge2$ works since we get that the ratio is $\le (1+1/2)\cdot {1\over 2}={3\over 4}$. Thus, the series converges.
As regards your tutor's expression, I don't get it either; as noted above, the correct expression would be $(1+1/n)\cdot {1\over 2}$, and $(n+1/n)\cdot {1\over 2}$ is in error.
$$\frac{n+1}{2^{n+1}}\cdot\frac{2^n}n =\frac{n+1}n\cdot\frac{2^n}{2^{n+1}}=\frac{n+1}n\cdot\frac{2^n}{2^n\cdot 2}=\frac{n+1}n\cdot\frac12.$$