power of matrix

by teomihai, Apr 30, 2025, 12:15 PM

Find $A^{n}$ ,where $A=\begin{pmatrix}1&-\frac{1}{5}-\frac{2}{5}i\\2+i&-i&\end{pmatrix}$
we now $i^2=-1$ ,and $n$ it is positiv integer number.

This post has been edited 2 times. Last edited by teomihai, Yesterday at 12:15 PM

linear algebra

matrix

Does the sequence log(1+sink)/k converge?

by tom-nowy, Apr 30, 2025, 10:35 AM

Does the sequence $\frac{\ln(1+\sin k)}{k} \;\;\;(k=1,2,3,\ldots)$ converge?

Evaluate: $\lim_{h\to 0^{-}} \frac{-1}{h}.$

by Vulch, Apr 30, 2025, 2:33 AM

Respected users,
I am asking for better solution of the following problem with excellent explanation.
Thank you!

Evaluate: $\lim_{h\to 0^{-}} \frac{-1}{h}.$

This post has been edited 2 times. Last edited by Vulch, Yesterday at 2:34 AM

calculus

integration

by We2592, Apr 25, 2025, 8:51 AM

Q) solve the integration
$\int_{0}^{\alpha} \frac{d\theta}{\sqrt{cos\theta-cos\alpha}}$

integration

calculus

real analysis

Putnam 1958 February A1

by sqrtX, Jul 18, 2022, 9:38 PM

If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying
$\frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$ show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.

Putnam

roots

polynomial

Putnam 1952 B1

by centslordm, May 30, 2022, 4:18 PM

A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A,$ to find the third side $a.$ He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c;$ subtracts the sum of the logarithms of $2, b, c,$ and $\cos A;$ divides the result by $2;$ and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?

Putnam

Putnam 1952 A1

by centslordm, May 29, 2022, 5:51 PM

Let $f(x) = \sum_{i=0}^{i=n} a_i x^{n - i}$ be a polynomial of degree $n$ with integral coefficients. If $a_0, a_n,$ and $f(1)$ are odd, prove that $f(x) = 0$ has no rational roots.

Putnam

Putnam 1951 B3

by centslordm, May 25, 2022, 10:48 PM

Show that if $x$ is positive, then $\log_e (1 + 1/x) > 1 / (1 + x).$

Putnam

Putnam 1980 B3

by sqrtX, Apr 1, 2022, 4:10 PM

For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$

Putnam

Sequences

Range of 2 parameters and Convergency of Improper Integral

by Kunihiko_Chikaya, Aug 21, 2012, 5:38 PM

Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.