Find the value of $$ \cot\left(\sum_{n=1}^{23} \cot^{-1} (1+\sum_{k=1}^n 2k )\right)$$
I tried to simplify it but it became messy. How can I find $\sum_{n=1}^{23} \cot^{-1} (1+n+n^2)$? Please help me.
2026-03-25 07:48:53.1774424933
Find $ \cot\left(\sum_{n=1}^{23} \cot^{-1} (1+\sum_{k=1}^n 2k )\right)$
104 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in TRIGONOMETRY
- Is there a trigonometric identity that implies the Riemann Hypothesis?
- Finding the value of cot 142.5°
- Using trigonometric identities to simply the following expression $\tan\frac{\pi}{5} + 2\tan\frac{2\pi}{5}+ 4\cot\frac{4\pi}{5}=\cot\frac{\pi}{5}$
- Derive the conditions $xy<1$ for $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}$ and $xy>-1$ for $\tan^{-1}x-\tan^{-1}y=\tan^{-1}\frac{x-y}{1+xy}$
- Sine of the sum of two solutions of $a\cos\theta + b \sin\theta = c$
- Tan of difference of two angles given as sum of sines and cosines
- Limit of $\sqrt x \sin(1/x)$ where $x$ approaches positive infinity
- $\int \ x\sqrt{1-x^2}\,dx$, by the substitution $x= \cos t$
- Why are extraneous solutions created here?
- I cannot solve this simple looking trigonometric question
Related Questions in SUMMATION
- Computing:$\sum_{n=0}^\infty\frac{3^n}{n!(n+3)}$
- Prove that $1+{1\over 1+{1\over 1+{1\over 1+{1\over 1+...}}}}=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}$
- Fourier series. Find the sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n+1}$
- Sigma (sum) Problem
- How to prove the inequality $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n-1}\geq \log (2)$?
- Double-exponential sum (maybe it telescopes?)
- Simplify $\prod_{k=1}^{l} \sum_{r=d}^m {{m}\choose{r}} \left(N-k \right)^{r} k^{m-r+1}$
- Sum of two martingales
- How can we prove that $e^{-jωn}$ converges at $0$ while n -> infinity?
- Interesting inequalities
Related Questions in INVERSE-FUNCTION
- Derive the conditions $xy<1$ for $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}$ and $xy>-1$ for $\tan^{-1}x-\tan^{-1}y=\tan^{-1}\frac{x-y}{1+xy}$
- Combination of functions and their inverses.
- Solve $\sin^{-1}x+\sin^{-1}(1-x)=\cos^{-1}x$ and avoid extra solutions while squaring
- Find the greatest and least values of $(\sin^{-1}x)^2+(\cos^{-1}x)^2$
- Is it always possible to rearrange an equation desirably?
- Only bijective mappings are invertible. Clarifying proof.
- Relating the roots of quadratic to an inverse trigonometric functions' question
- Derive the conditions for $\tan^{-1}\frac{a\cos x-b\sin x}{b\cos x+a\sin x}=\tan^{-1}\frac{a}{b}-x$
- Why is the inverse of the derivative of f not the actual derivative of the inverse of f?
- $\ {\sin}^{-1}{(x)} $ equation and function
Related Questions in TELESCOPIC-SERIES
- Double-exponential sum (maybe it telescopes?)
- Calculate the sum of the series: $\sum\limits_{n=1}^{\infty}\left ( \frac{3}{n^{2}+n}-\frac{2}{4n^{2}+16n+15} \right )$
- Squared Summation(Intermediate Step)
- Another solution.
- I am stuck on a question Algebra:Sequence and series
- What is the closed form of the sequence : $ a_{n+2} = 2a_{n+1}-a_n+2^n+2 $
- Find the closed form of $u_{n+1}=a_nu_n+b_n$
- Want to understand how a fraction is simplified
- Evaluate a sum which almost looks telescoping but not quite:$\sum_{k=2}^n \frac{1}{k(k+2)}$
- Compute the values of the Double Sum
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?
The hint.
Use $$\arctan(n+1)-\arctan{n}=arccot(n^2+n+1)$$ and the telescopic summation.