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Can we always conclude limit superior and limit inferior of $\langle a_n\rangle=\langle b_nc_n\rangle$?

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#real-analysis #sequences-and-series #cauchy-sequences #limsup-and-liminf
151
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Prove or disprove $\sum_{k=0}^\infty\|y_{k+1}-y_k\|<\infty$ if $y_{k+1}=\big(I_n-\frac{ax_kx_k^{\rm T}}{b+\|x_k\|^2}\big)y_k$

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#real-analysis #linear-algebra #sequences-and-series #matrices #vector-analysis
46
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Find the limit of $d_1 = 1, d_{n+1}=1+\frac{d_n}{3}$

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#real-analysis #sequences-and-series #limits #proof-verification #proof-writing
51
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Infinite real number serie

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#calculus #sequences-and-series
58
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Finding the sum of a recursively defined function

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#sequences-and-series
60
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Prove that the given series $\sum_{(m,n)∈\mathbb{Z^2})} |\max\{m,n\}|^{-5}$ is convergent.

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#sequences-and-series
215
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$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{\infty}=?$

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#sequences-and-series #geometric-progressions
1.1k
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Evaluate $\lim_{n\rightarrow\infty}\sum_{k=1}^n\arcsin(\frac k{n^2})$

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#sequences-and-series #limits #definite-integrals #summation #limits-without-lhopital
85
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Sum of an arithmetic series

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#sequences-and-series
150
Views

The condition of convergence for series $\sum_{n=0}^{\infty} x^2_n$

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#sequences-and-series #functional-analysis #convergence-divergence
140
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Limit of a sum with constant

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#real-analysis #sequences-and-series #limits #convergence-divergence
75
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Finite Sum of Power of Cosine Function

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#real-analysis #sequences-and-series #trigonometry #summation
32
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Relationship between divergence of these two series

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#sequences-and-series #convergence-divergence #divergent-series
64
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Convergent sequences to the same limit

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#sequences-and-series
65
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Show that $\{\frac{e^n}{n!}\}$ diverge.

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#sequences-and-series

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