I'm new to real analysis and trying to solve a basic questions. I'm asked to prove that the following sequence is monotonically decreasing: $b_{n}=\sum_{k=1}^n\frac{(-1)^{2k}}{2k+1}$. So I need to prove $b_{n} \leq b_{n+1}$ Now I think I should use induction.
basisstep: $b_{1}=\frac{1}{3}$.
induction hypothesis: assume $b_{n+1} \geq b_{n}$.
Proof. Since $b_{n+1}=b_{n}+\frac{(-1)^{2n+2}}{2n+3}=b_{n}+\frac{1}{2n+3}$. And observe that $\frac{1}{2n+3}>0$. So it follows that $b_{n+1}=b_{n}+\frac{1}{2n+3}>b_{n}$.
Is this a valid proof or am I missing something?
Thanks in advance.
UPDATE First of all, thanks for the fast comments. I now realise I misinterpreted the proof. Instead of proving that the summation is monotonically decreasing, I need to prove that the elements consisting of (1/3,1/5,1/7..1/(2n+1) is monotonically decreasing.
simplify your calculations and observe that $$(-1)^{2k}=1$$ for all $k$