Separately, I can show that both parts of the sum diverge, but that doesn't help me very much. I don't see any test which could come in handy. I know both sides of the sum have popular Taylor forms, so I can partially represent them as such around $x=0$, but my question then what do I do with the remainder?
2026-04-04 17:34:07.1775324047
Showing that $\sum^\infty_{n=1}\left(\sin\left(\frac{1}{n}\right)-\ln\left(1+\frac{1}{\sqrt n}\right)\right)$ diverges
57 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in SEQUENCES-AND-SERIES
- How to show that $k < m_1+2$?
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Negative Countdown
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Show that the sequence is bounded below 3
- A particular exercise on convergence of recursive sequence
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Powers of a simple matrix and Catalan numbers
- Convergence of a rational sequence to a irrational limit
- studying the convergence of a series:
Related Questions in TAYLOR-EXPANSION
- Mc Laurin and his derivative.
- Maclaurin polynomial estimating $\sin 15°$
- why can we expand an expandable function for infinite?
- Solving a limit of $\frac{\ln(x)}{x-1}$ with taylor expansion
- How to I find the Taylor series of $\ln {\frac{|1-x|}{1+x^2}}$?
- Proving the binomial series for all real (complex) n using Taylor series
- Taylor series of multivariable functions problem
- Taylor series of $\frac{\cosh(t)-1}{\sinh(t)}$
- The dimension of formal series modulo $\sin(x)$
- Finding Sum of First Terms
Related Questions in DIVERGENT-SERIES
- Proving that a series is divergent?
- Is this : $\sum_{n=1}^{+\infty}\frac{(-1)^n}{\tan(n!)}$ convergent sum?
- Convergence (or Divergence) of [(-1)^(n-1)*e^(1/n)]/n
- Show that $\sum_{n=1}^\infty \frac{1}{(\log(n))^p}$ diverges
- Showing Harmonic series is divergent. (question on summation properties)
- Testing convergence of $\sum\limits_{n=1}^{\infty}u_n$ , where $u_n = \left ( 4- \frac{1}{n}\right) ^ { \frac{( - 1) ^ {n }}{ n}}$
- how to prove $\sum \frac {|\alpha+\sin(n^2)|}n$ diverges without summation by parts?
- Behavior of a sum on the boundary of convergence/divergence
- Divergence of a Series $\sum_{n=1}^\infty (\frac{1}{n!})(\frac{n}{e})^n$
- $\sum_{n=1}^{\infty} n $ equals to another value than $\frac1{12}$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
$$\begin{split}\sin\left(\frac 1 n\right)-\ln\left(1+\frac 1 {\sqrt n}\right)&=\left[\frac 1 n +\mathcal O\left(\frac 1 {n^3}\right)\right]-\left[\frac 1 {\sqrt n}-\frac 1 {2n}+\mathcal O\left(\frac 1 {n\sqrt n}\right)\right]\\ &=-\frac 1 {\sqrt n}+\frac 3 {2n}+\mathcal O\left(\frac 1 {n\sqrt n}\right)\\ &\sim -\frac 1 {\sqrt n}\end{split}$$