1. A

    Prove the series are not Cauchy using the Cauchy criterion with epsilon

    $b_n=\frac{x}{1}+\frac{x+1}{3}+...+\frac{x+n}{2n+1},x>1$ $c_n=\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+...+\frac{1}{\sqrt n},n≥1$ I'm having troubles with the Cauchy criterion and I do not understand how to apply it when the series are not convergent. These exercises are telling me to use the Cauchy...
  2. W

    absolutely convergent series

    Hello. I want to show that, for an absolutely convergent series \sum_{n=1}^{\infty}a_n, we have \left|\sum_{n=1}^{\infty}a_n\right|\leq\sum_{n=1}^{\infty}|a_n|. Let M be an positive integer. I begin with the triangle inequality \left|\sum_{n=1}^{M}a_n\right|\leq\sum_{n=1}^{M}|a_n| and taking...
  3. tahirimanov19

    Generalization of Harmonic Series

    $h(x)$ is a generalized $H_n$, and $H_n = \sum\limits_{k=1}^n \dfrac{1}{k}$. Assumptions: For $n \in \mathbb{N}$, $h(n)=H_n$; $h(1)=1$; $h(x)=h(x-1) + \dfrac{1}{x}$. I started by $$\Large{f(n)= \int_1^n h(z)dz}$$ and at next page I derived at $$\Large{\int_1^n h(z) dz = \ln (n!) +...
  4. tahirimanov19

    Geometry Problem Series, Question 3:

    Pentagon $ABCDE$ is inscribed in a circle. $AB \parallel EC$, $AE \parallel BD$. $AD \cap EC \equiv G$, $BD \cap EC \equiv F$ and $AC \cap BD \equiv H$. Prove that the area of $AGFH$ is equal to the sum of the areas of $DEG$ and $BCH$. --------
  5. tahirimanov19

    Geometry Problem Series, Question 2:

    $T$ is the middle point of the segment $AB$ of the convex quadrilateral $ABCD$. The circle $\omega$, through points $C,D,T$, is tangent to $AB$. $K$ and $L$ are the intersection points of $AD$ and $BC$ respectively with $\omega$. $M$ and $N$ are the intersection points of $AC$ and $BD$...
  6. tahirimanov19

    Geometry Problem Series, Question 1:

    A circle circumscribed to the triangle $ABC$. $D$ is a point on the circumcircle. Prove that symmetries of the point D according to the edges of the triangle are collinear.
  7. I

    Cannot understand harmonic series logarithmic growth

    I know that 1+1/2+1/3+...+1/n =\ln(n)+\gamma+\epsilon_{k}=f(n). where \epsilon_{k}\approx 1/2n. The problem is this sum : 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n^2 }=? , can we go like H_{n^2 } =f(n^2 )? If yes then (*) \: \displaystyle 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n^2...
  8. A

    Help series

    Please help me with the attached problem!
  9. H

    Word Problem Sequences and Series

    A Mining Company was founded in 1894 and the mine’s initial production was the extraction of 100kg/year of silver. Each following year saw a steady increase of 60kg/year until the silver production peaked at 700 kg/year. Production remained at this level until 1914, when an event caused the...
  10. I

    Harmonic series

    Find the average value of H_n =1+1/2+...+1/n.
  11. A

    Cauchy series criterion

    I have two exercises: $$b_n=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}$$ $$n\geq1$$ $$c_n=\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+...+\frac{1}{\sqrt{n}}$$ $$n\geq1$$ Do I replace the terms $x_{n+p}$ and $x_n$ with: \left| \frac{1}{(n+p)^2}-\frac{1}{n^2} \right| \lt ε \left|...
  12. A

    Two positive series with terms in a fraction

    $$\mathop{\mathrm Σ}_{n=1}^\infty \frac{1+1/2+...+1/n}{n}$$ I have two series in a fraction and I do not understand how to solve this problem.I see that the numerator is a Harmonic series but that doesn't help me a lot.I tried doing the comparison test and I could only compare this series to...
  13. A

    Proof for square root series In this image I have a sum series and I need to proove that all the terms from the LHS are equal to the two terms from the RHS. I wrote a solution but I'm not 100% certain this is correct.
  14. K

    Convergence of mean of inf series of random variables with different distributions

    Please see the attachment figure. I have an arithmetic mean of an infinite series of independent random variables. However, these variables can come from 5 different independent normal distributions, and each of the 5 distributions are equally probable (each having its own mean and standard...
  15. M

    Infinite series

    hi I found this infinite series in my calculations and I want to ask if it's have a name . I attached it to this thread. if anyone here knows anything about it ,please contact me.

    Infinite series

    I cannot remember the infinite series for \sqrt {1 - a} for 0 < a <<1
  17. A

    Prove the limit of natural logarithm without differentiation or Taylor series

    Can someone help me to prove that this limit is equal to 1 without using differentiation or Taylor Series?(like in the limit of sinx/x when x tends to 0)What are the steps?
  18. M

    Definition of Formal power series in m indeterminates over R

    I could not understrand the following definition for formal power series over $m$ indeterminates, over the commutative ring $R$: *I do understand:* We set $R[\![X_{1},...,X_{m}]\!]:=(R^{(\mathbb{N}^{m})},+,.)$, where $+$ and $.$ are as in: $(p+q)_{\alpha}:=p_{\alpha}+q_{\alpha}$...
  19. R

    Expression for finding common elements in two series

    Hello, First of, I am sorry if I am posting it in a wrong section. If so, can someone please move this thread to the appropriate section? Now, my problem: I have a series of the form S1=x(x+1) and another series S1/k, for any k∈N. Now I want to find the values where the elements of two series...
  20. M

    Limit of a series

    Does anyone know how to solve this limit? $$\lim_{n\to\infty} \sum_{k=0}^{n-1} \Big(x+\frac{1}{n}\Big)^k $$ for 0<x<1 $\\$ Thanks!