The -th sum is:
So, I am pretty sure you cannot prove that sum is less than or equal to .
I see what you are saying...
Let's try to see when we add term more than once: For each , .
Some examples:
So,
Now, the n-th partial sum is and
This still isn't quite right, but it might give you some ideas. Now, I am not adding 1/4^4 at all.
Look at estimation of error.
Since is an alternating sum (similar to a telescoping sum), it is no bigger than its first term, which is . This part should not be difficult to prove.