sequences-and-series

  1. Expected value sum identity

  2. The doubly infinite series $\sum_{-\infty}^{+\infty} n$

  3. limit of a recursive sequence, then show how it can be unbounded and not monotone
  4. Convergence of a random created series

  5. The set of functions which map convergent series to convergent series
  6. Find the number of bacteria after dividing

  7. Convergence of $\sum_{n\geq1}\frac{\sin^n(n)}{n}$
  8. How can I study the convergence of $\sum_{n=1}^∞ \frac{n!}{n^n}$ using the comparison test?
  9. Related divergent series
  10. Find a p-series to compare to the given one to determine convergence or divergence.

  11. How to find a general formula for any number sequence? $19,25,45,87,159$
  12. How to find the initial terms of the recurrence relation if you know the nth term?
  13. Whats is the sum of series for each $0\le x\le 3$?

  14. Find the sum of the following series. $\sum_{m=1}^{ \infty }\frac{{(-1)^m}{+3}}{5^m}$

  15. Does $\lim\limits_{x\to 1^{-}}\sum_{n=1}^{\infty}\frac{x^n}{n(n+1)}$ exist?

  16. Proving that this sequence is convergent

  17. For what value of $\alpha$, does this series converge?

  18. If $\sum a_n$ and $\sum b_n$ and their convolution $\sum c_n$ converge, then $\sum c_n=\sum a_n\sum b_n$

  19. Evaluate $\int_{-\pi}^\pi \big|\sum^\infty_{n=1} \frac{1}{2^n} e^{inx}\big|^2 \operatorname d\!x$

  20. What's wrong with asking for the next term in a sequence?

  21. Does $\operatorname EX_n\to0$ as $n\to\infty$?

  22. Are the sets $\mathbb X=\{0,1,4,15,56,...,x_h,...\} $ and $\mathbb Y=\{0,2,12,70,408,...,y_i,...\}$ (excepting the elements $x_0=y_0=0$) disjunct?
  23. Implications of $ \lim_{n\rightarrow \infty}\Big[ nP_j-\frac{\exp(x_j)}{1+\frac{1}{n}\sum_{k=1}^n \exp(x_k)}\Big]=0 $ $\forall j\in \mathbb{N}$

  24. Convergence of a Polynomial Series

  25. Determine whether the following series converges absolutely , conditionally or diverges:

  26. If AM-GM holds for any $2^n$ positive real numbers, show that AM-GM holds for *any number* of positive real numbers.

  27. Mathematical induction for Fibonacci

  28. Just another infinite resistor ladder - question

  29. Is the sum of digits of $\left(16^k - 1\right)$ less than $6k$ for $k > 223$?
  30. Computing Fourier Series and it's modes of convergence.
  31. Prove or disprove if $x_k$ is unbounded and $|P|\geq 0$, then either $\lim_{k\to\infty}x_k^{\rm T}Px_k=0$ or $x_k^{\rm T}Px_k$ is unbounded
  32. Calculating the summation $\sum_{k=-\infty}^{\infty}\frac{e^{ika}}{1+|k|^r}$

  33. Proving sequence of functions has no convergent subsequence

  34. Determine all limits of subsequences of $|\lambda \alpha^n +\mu \bar{\alpha}^n|$ (assume $|\alpha|>1$)

  35. Does the series $\sum_{n=1}^{\infty} \frac{\tan(2n+1)}{(2+n)^{1/2}}$ converge absolutely?

  36. $\sum_{n=0}^{\infty} \frac{(-1)^n}{1+\sqrt n}$
  37. Can one differentiate an infinite sum?
  38. How can I calculate the mean number of attempts needed for success?
  39. Finding the limit of $\left( 1-\frac{1}{n} \right)^{n}$
  40. If $\frac{a_n}{b_n} \rightarrow 1$ does $a_n$ converge?

  41. Determine convergence or divergence of the following series
  42. $x_1=1,x_n=x_{n+1}+\ln (1+x_{n+1})$, prove $x_n\leq\frac{1}{2^{n-2}}$.
  43. $\sum_\limits{n=1}^{\infty}\frac{1}{n!x^n}$ and $\sum_\limits{n=1}^{\infty}\frac{1}{(2n-1)x^n}$ different convergence domains?

  44. How do I prove that a limit exists for $x(n)= 1/(n+1) + 1/(n+2)+\cdots+1/(n+n)$?

  45. Does $\{(-1)^n\mid n\in\mathbb N\}=\{-1,1\}$
  46. Asymptotic expansion of $u_n$ such that $u_n^n-u_n-n=0$

  47. Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?
  48. Uniform Convergence of Dirichlet Series over $\mathbb{R}$

  49. Is there a closed form for the sequence $1, 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, ... $?
  50. How do I find the sum of the series -1^2-2^2+3^2+4^2-5^2… upto 4n terms?

  51. $I(x_1, x_2, . . . , x_6) = 2x_1 + 4x_2 + 6x_3 + 8x_4 + 10x_5 + 12x_6 \mod 10$ is the invariant
  52. Question on Convergence of sequence
  53. What is the next term of the series 4,6,8,10,20,-6,70,30,350,-34,2450...?

  54. cauchy sequences and their relation to sums of distances in metric spaces
  55. Given the sum of this series, find the sum of another series.

  56. Find sequence $(a_n)$ and $(b_n)$

  57. Complete series expansion of $\frac{1}{1+f(x)}$
  58. series question to use for mathematical induction

  59. Positive sequence $\{a_n\}$, $a_{n+1}=-\sqrt{a_n}+1, a_1\in{(\frac12,1)}$. Prove $a_{2n+1}+a_{2n+2}<a_{2n-1}+a_{2n}$.
  60. The integral $\int_0^{\frac{\pi}{2}}\left(\frac{x}{\sin x}\right)^2\text{d}x$
  61. Proof of $(1+x)^\alpha$ converges to its Taylor expansion at $x_0=0$ when $-1<\alpha<0$ and $x=1$

  62. Convergence of Series with Ratio Test
  63. How to calculate $\sum_{k=1}^{\infty}x^{k^2}$ for different $x$
  64. Uniform convergence of series based on the Weierstrass $M$-test

  65. Prove that $\sum_k\sin(2 k \arctan(k^2))$ converges

  66. Finding the Maclaurin series of a given function knowing another Maclaurin series

  67. Show that the sequence $\sum\limits_{k=1}^n\cos\left(\frac kn\right)^{2n^2/k}$ converges

  68. Looking for a method to maximize a sum
  69. Does $\sum _{n=1}^{\infty }\left(a_n\:+\:b_n\right)^2$ converges?
  70. Prove common ratio of GP from a equation
  71. Modified alternating harmonic series

  72. When is it necessary to increase $N$ when $\epsilon$ decreases for Cauchy sequences?
  73. $\{a_n\}_{n\ge 1}$ be a sequence of non-zero real numbers, is there a subsequence $\{b_n\}_{n\ge 1}$ s.t. $b_{n+1}/b_n \to 0,1$ or $ \infty$?

  74. Does Fourier cosine series converge in $L^2$ sense?
  75. Summation of this series.
  76. Nature of the series $\sum\limits_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$

  77. Test for convergence of the series $\sum\limits_{n=3}^\infty\frac{1}{(\ln \ln n)^{\ln n}}$

  78. Help with the application of the infinite geometric series rule.
  79. Q: Sequence definition of non-convergence (Not diverging definition)
  80. Simplifying a sum involving Stirling numbers of the second kind
  81. Find the sequence given its generating function

  82. Inequality for Fibonacci to find an upper bound of harmonic Fibonacci series

  83. Calculate the following convergent series: $\sum _{n=1}^{\infty }\:\frac{1}{n\left(n+3\right)}$
  84. if $0$ is not an accumulation point for $\{1/b_n\}$, then $\{b_n\}$ converges to $b$.

  85. Initial values of Inkeri's primality test for Fermat numbers

  86. A conjecture on bounded complex partial sums

  87. Convergence in probability of this sum

  88. Integral Representation of the Zeta Function: $\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$
  89. Show that $\sum\limits_{1\leq m^2+n^2 \leq R^2} \frac{1}{m^2+n^2} = 2\pi \log R +O(1)$ as $R\to \infty$

  90. Is the sequence $\{\varphi^{(k)}(n)\}$ an eventually periodic sequence?

  91. Sum of reciprocals of the square roots of the first N Natural Numbers

  92. Does $\sum_{n=1}^{\infty}\frac{\cos^{2}(n+1)}{n}$ converge?
  93. Existence of sequence of non-atoms and questions on sequences and notations
  94. Extending Sophomore's Dream to include a constant in the exponent.

  95. What is the value of $\sin 1 ^\circ \sin3^\circ\sin5^\circ \sin 7^\circ \sin 9^\circ \cdots \sin 179^\circ $?
  96. Given a power series with interval of convergence $(-1,1]$, construct a series with another given interval of convergence
  97. Sum of a particular arithmetic sequence

  98. How do I express this proof of absolute convergence as tersely as possible?

  99. Does uniform convergence imply convergence in measure?
  100. Determine if the following series converges or diverges: $\sum _{n=1}^{\infty }\ln\left(\frac{n}{n+1}\right)$