How to test the convergency of a series How to test that the following series is convergent $$\frac{1}{2}+\frac{1}{2+1}+\frac{1}{2^2+1}+\frac{1}{2^3+1}+\dots$$ attempt: $$\lim_{n\to ∞}\frac{u_{n+1}}{u_n}=\lim_{n\to ∞}\frac{2^n+1}{2^{n+1}+1}$$ Then how to get the value. Is it $\frac{1}{2}$?

$$\lim_{n\to ∞}\frac{u_{n+1}}{u_n}=\lim_{n\to ∞}\frac{2^n+1}{2^{n+1}+1}$$ divide the Numerator and denominator by $2^{n+1}$ $$\lim_{n\to ∞}\frac{\frac{2^n}{2^{n+1}}+\frac{1}{2^{n+1}}}{\frac{2^{n+1}}{2^{n+1}}+\frac{1}{2^{n+1}}}=\lim_{n\to ∞}\frac{\frac{2^n}{2^{n}.2}+\frac{1}{2^{n+1}}}{\frac{2^{n+1}}{2^{n+1}}+\frac{1}{2^{n+1}}}=\frac{1/2+0}{1+0}=0.5$$