Prove sequences are both convergent, given $ a_{n+1} = \sqrt{a_{n}b_{n}} $ and $ b_{n+1} = \frac{a_{n} + b_{n}}{2} $
2799
Question about arithmetic–geometric mean
122413
Show $b_n=\sqrt[k]{n+1}-\sqrt[k]{n}$ converges towards $0$ for all $k \geq 2$.
101766
Recursive sequence convergence.: $s_{n+1}=\frac{1}{2} (s_n+s_{n-1})$ for $n\geq 2$, where $s_1>s_2>0$
116414
Proving that the difference of the terms of a sequence, multiplied by $n$ converge
78127
Convergence of non increasing and non decreasing sequences to the same limit
35885
Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$
34025
Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$
46685
Show that the sequence $a_1=1$, $a_2=2$, $a_{n+2} = (a_{n+1}+a_n)/2$ converges by showing it is Cauchy
35731
Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$.
60036
How to prove this limit exists: $a_{n+2}=\sqrt{a_{n+1}}+\sqrt{a_{n}}$
105882
Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$
16970
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
859
If $\sum a_nb_n$ converges for each null sequence $(b_n)$ then $\sum a_n$ is absolutely convergent
124615
Proving convergence of sums: If $\sum_{n=1}^{\infty}a^2_n$ and $\sum_{n=1}^{\infty}b^2_n$ converge then $\sum_{n=1}^{\infty}a_nb_n$ converges
116046
Prove that if $\sum_{n=1}^\infty a_{2n}$ and $\sum_{n=1}^\infty a_{2n-1}$ converge then $\sum_{n=1}^\infty a_n$ converges.
121774
Prove that $ \left(a_{n}\right)_{n=1}^{\infty} $ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $ 0<q<1 $
50087
Convergence of $a_{n+1}=\frac{1}{1+a_n}$