sequences-and-series

  1. convergence of a nonnegative monotonic sequence

  2. Equivalent Sequential Definitions of Continuity

  3. A problem on sequence of real numbers
  4. What is the domain of convergence for $ \sum_{(a,b) \in \mathbb{Z}^2} \frac{a^4 + b^4}{(a^2 + b^2)^s} $ with $(a,b) \neq (0,0)$?
  5. Pi series that converges arbitrarily fast.

  6. Show that either there exists an $N$ so that $r_n=r$ for $n \ge N,$ or the set of numbers $\{q_n\}$ is unbounded.
  7. Second digit of square numbers in binary yields $\sqrt2$

  8. Interest rates are an example of a geometric sequence

  9. Show that the sequence $a_n = \frac{(n+1)^2 -n^2}{n}$ converges and give its limit.

  10. Comparative status of $\sum a_n$ and $\sum {a_n}^{[2]}$.
  11. How to find $\sum \left(1 + 1/2 + \dots + 1/(n + k + 1)\right)\frac{1}{n(n + 1)...(n + k + 1)}$?

  12. Every integer valued Cauchy sequence is ultimately constant

  13. Check the convergence of a series
  14. Find floor of sum $\sum_{k=1}^{80} k^{-1/2}$

  15. Finding the sum of arithmetic series when last term and common difference is given .

  16. For all polyomial $P$ in $\mathbb C[X]$ there exists a norm such that $(X^n)$ tends to $P$

  17. Showing Convergence in a Set of Sequences

  18. Differentiation and Uniform Convergence

  19. Upper hemicontinuity and closed graphs
  20. Summation of n terms of series using calculus
  21. Special $\zeta$-series including primes.

  22. Are there sequences with the property: If $\sum^{\infty}_{n=0} a_{n}=s$ then $\sum^{\infty}_{n=0} 1/a_{n}=1/s$

  23. Probability of a sequence of playing cards in a specific order

  24. Let $(a_n)$ be a sequence of positive real numbers such that $\sum\limits_{n=1}^{\infty} a_n$ is convergent.
  25. How do I find the Number of ways to reach n− n, without the score having been tied before?
  26. Sum of logarithmic series given the sum of coefficients

  27. How to prove the convergence of the sеquence?

  28. A claiming series to be of rational values
  29. Showing the existence of subsequences
  30. Name of $\prod_{n = 1}^{\infty}n = 1 \times 2 \times 3 \times 4 \times 5 \times \cdots$
  31. Relation between number of digits and term index in look-and-say sequence.
  32. Interesting sums yield rational value

  33. An example of a sequence $(a_n)_{n \in \mathbb{N}}$

  34. What are some mathematically interesting computations involving matrices?
  35. Manipulation of factorials
  36. Why is this series divergent?

  37. limit of $\frac{(2n)!}{4^n(n!)^2}$

  38. Prove: If $x_n=o(\alpha_n)$, then $x_n=O(\alpha_n)$. Show that the converse is not true.

  39. Distance from center of a spiral
  40. An example of a sequence
  41. Smallest $\lambda$ such that $\sum_{n=1}^{\infty} \frac{n}{\sum_{k=1}^{n}a_k} \le \lambda \sum_{n=1}^{\infty} \frac{1}{a_n}$

  42. Magic of sums of $5$th and $7$th powers of natural numbers: $\sum_{i=1}^n i^5+i^7=2\left( \sum_{i=1}^ni\right)^4$?

  43. Summing over trivial zeros of Riemann zeta function in explicit formulas

  44. Show that $\sum_{k=2}^\infty d_k$ converges to $\lim_{n\to\infty} s_{nn}$.
  45. Cauchy Product of Two Divergent Series

  46. If $f(x) = 0$ for all irrational and $x_n$ is a sequence of irrationals, is $\lim_{n\to\infty} f(x_n)= 0$?
  47. Prove convergence / divergence of $\sum_{n=2}^\infty(-1)^n\frac {\sqrt n}{(-1)^n+\sqrt n}\sin\left(\frac {1}{\sqrt n}\right)$

  48. Find $A^n$, if $n$ is Natural number
  49. The convergence of the series $\frac{1}{n(\ln(n))^2}$ depends on the starting index?
  50. Looking for a proof of an interesting identity

  51. Optimal probability mass function with domain $\mathbb N^+$

  52. Expansion of The Reciprocal of a Bessel Function

  53. Proof by induction of a sum - how can I complete the proof?
  54. Convergence or divergence of $a_n=\frac{(n-1)!^2 \cdot x^{2n-2}}{(2n-2)!}$
  55. Limit of sequence in which each term is defined by the average of preceding two terms
  56. Series of Gamma functions involving $\Gamma \left(\frac{n}{2} (1-i x)\right) \Gamma \left(\frac{n}{2} (1+i x)\right)$?

  57. Is it true that $\sum_k a_k(n)\to0$ as $n\to\infty$ if $a_k(n)\to0$ for all $k$ and $\sum_k a_k(n)$ converges for all $n$?

  58. Show that $\lim x_n$ exists.

  59. Convergence of series $(\frac{1}{3})^{2}+(\frac{1.2}{3.5})^{2}+(\frac{1.2.3}{3.5.7})^{2}+...$

  60. Picard Iteration: Convergence of system

  61. Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)
  62. square summable and summable

  63. A probabilistic average integral

  64. How can I determine if a number is a part of arithmetic sequence
  65. Prob. 4, Chap. 7, in Baby Rudin: On what intervals does the series converge uniformly?
  66. $a_n$ does not have an upper bound$\iff a_n$ has a subsequence which diverges to infinity

  67. Convergence with even and odd subsequences

  68. Best objective function for maximize the distance between a group of series

  69. Find the general term of the sequence, starting with n=1
  70. Non-Equidistant Sequences

  71. Can a sequence of functions diverge to infinity in one point and converge a finite limit in another?

  72. 16 digit number with only 1 and 2s
  73. How to prove that: $\lim_{n\to\infty}\frac1{n!}\int_0^ne^{-t}t^n\,dt.=\frac{1}{2}$

  74. Show that, for $n\in \mathbb{N}$, the sequence is converging.

  75. Solving recurrence relation with square root.

  76. How do I go about solving various summation of binomial coefficients like $\sum_{r=0}^{n} \binom{n}{r}f(r)$
  77. Analyze the convergence of the series: $\sum_{n=1}^{\infty} \frac{n^n}{(n+1)!}$

  78. infinite series,comparision test,convergence divergence
  79. What's the closed form of this :$\sum_{n=1}^{+\infty}\frac{(-1)^n\phi(n)}{n}$
  80. Why is the following statement true in a proof that I'm studying on Binary Search Trees?

  81. Infinite Sum Proof

  82. What is the 1000th decimal of the square root of 1998 ones?

  83. Is it divergent? $\lim_\limits{n\to\infty}\frac{n!^{3}}{(3n)!}$

  84. Verification of infinite square root equation proof
  85. Closed form of an infinite sum $\sum_{k=1}^\infty\frac{1}{k(k+p)}$
  86. A series of nonanalytic-smooth functions $f'_n = f_{n+1}$ with finite sum?

  87. Sandwich/Squeeze Theorem for Null Sequences

  88. Set values of $x \in \mathbb{R}$ in a way that given series is convergent.

  89. Putting a particular sequence of sets in closed form

  90. Prove that $\sum a_k$ convergent implies $ka_k \to 0.$
  91. How to find $\lim\limits_{n\to +\infty}(3+\frac{1}{a_n})$, where $a_n = 3+\frac{1}{a_{n-1}}$?

  92. Determine the common difference for the arithmetic series given $s_n$ and $a$

  93. Finding the limit in a Fourier Series
  94. How to prove that $\sum_{n=1}^{\infty}\frac{\left ( -1 \right )^{n+1}n}{n^{2}+x^{2}}$ is always positive for all real $x$?

  95. If$ (X,d)$ is the discrete metric space and that $(x_n)_{n\in\mathbb N}$ converges to $x$ in it

  96. Show that if $x_n=O(\alpha_n)$, then $x_n/\ln n=o(\alpha_n)$
  97. Evaluating $\sum_{k=0}^\infty \left(\frac{1}{5k+1} - \frac{1}{5k+2} - \frac{1}{5k+3} + \frac{1}{5k+4} \right)$
  98. proving a formula about Arithmetic sequence

  99. How to calculate the series $\sum_{n=0}^{\infty}\frac{1}{1+\left(\frac{b+2bn}{a}\right)^{2}}$?
  100. Find the number of terms in the geometric series, given the first and the third term and the sum