Question

Determine whether it is convergent or divergent
1/4 + 3/8 + 5/16 + 7/32 + ........ Please explain in debt, i don't understand it at very well.

Answer 1

Each term's numerator is odd, and each term's denominator is a multiple of 4.

So, you can write the sum of terms as an infinite series:
\[\frac{1}{4}+\frac{3}{8}+\frac{5}{16}+\frac{7}{32}+\cdots=\sum_{n=0}^\infty\frac{2n+1}{4(n+1)}\]
To test for convergence/divergence, I'd recommend first checking
\[\lim_{n\to\infty}\frac{2n+1}{4(n+1)}\]
If the limit is non-zero then the series diverges. If the limit is 0, you'll have to try another test.

Answer 2

for that infinite series
I came up with 4(2^n)

Answer 3

for the denominator

Answer 4

@zpupster, thanks for pointing out my error. I must have imagined seeing all multiples of 4.

The series should in fact be \(\displaystyle\sum_{n=0}^\infty\frac{2n+1}{4(2^n)}\), or \(\displaystyle\sum_{n=0}^\infty\frac{2n+1}{2^{n+2}}\).

Now, the limit test I mentioned earlier won't work. Ratio test should, though.