\(\displaystyle{\int_{{-\infty}}^{{\infty}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={1}\)

Tiefdruckot

Answered 2022-01-03
Author has **2049** answers

Vasquez

Answered 2022-01-08
Author has **8850** answers

Here are two well-known infinite series whose sum is 1:

\(\sum_{n=1}^{\infty} \frac{1}{2^n}=\frac12+\frac14+\frac18+\frac{1}{16}+\frac{1}{32}+\ldots=1\)

\(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\frac{1}{5 \cdot 6}+\ldots=1\)

The following series is less known, interesting although:

\(\sum_{n=1}^{\infty} \frac{1}{s_n}=\frac12+\frac13+\frac17+\frac{1}{43}+\frac{1}{1807}+\ldots=1\)

where denominators form Sylvester sequence: every term is equal to the product of all previous terms, plus one. For example

2

3=2+1

7=2 * 3+1

43=2 * 3 * 7+1

1807=2 * 3 * 7 * 43+1

and so on

asked 2021-12-06

How to derive trig identity from the common ones?

\(\cos^{2}x=\frac{1+\cos2x}{2}\)

asked 2022-01-03

Can you please show me the process that determines \(\displaystyle{\int_{{-{1}}}^{{1}}}{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{x}^{{2}}+{1}}}}}}=\pi\)? Unfortunately I only know that the derivative of \(\displaystyle{{\sin}^{{-{1}}}{\left({x}\right)}}\) is equal to the integrand of that, but I don't actually know what to do with it.

asked 2021-08-11

Use one or more of the six sum and difference identities to verify the identity.

\(\displaystyle{\tan{{\left({\frac{{\pi}}{{{4}}}}-\theta\right)}}}={\frac{{{\cos{\theta}}-{\sin{\theta}}}}{{{\cos{\theta}}+{\sin{\theta}}}}}\)

\(\displaystyle{\tan{{\left({\frac{{\pi}}{{{4}}}}-\theta\right)}}}={\frac{{{\cos{\theta}}-{\sin{\theta}}}}{{{\cos{\theta}}+{\sin{\theta}}}}}\)

asked 2021-12-18

Verify the identity:

a) \(\displaystyle{\cos{{2}}}{x}={{\cos}^{{{2}}}{x}}-{{\sin}^{{{2}}}{x}}\)

b) \(\displaystyle{\cos{{2}}}{x}={1}-{2}{{\sin}^{{{2}}}{x}}\)

a) \(\displaystyle{\cos{{2}}}{x}={{\cos}^{{{2}}}{x}}-{{\sin}^{{{2}}}{x}}\)

b) \(\displaystyle{\cos{{2}}}{x}={1}-{2}{{\sin}^{{{2}}}{x}}\)

asked 2021-08-13

Simplify the trigonometric identity. There should be no division signs in the answer.

\(\displaystyle{\frac{{{\cos{\theta}}+{1}}}{{{1}+{\sec{\theta}}}}}\)

\(\displaystyle{\frac{{{\cos{\theta}}+{1}}}{{{1}+{\sec{\theta}}}}}\)

asked 2021-12-08

I have a question thatasks me to find an algebraic expression for \(\displaystyle{\sin{{\left({\arccos{{\left({x}\right)}}}\right)}}}\). From the lone example in the book I seen they're doing some multistep thing with the identities, but I'm just not even sure where to start here. It's supposed to be \(\displaystyle\sqrt{{{1}-{x}^{{2}}}}\)

asked 2021-12-19

How do you verify the identity \(\displaystyle{\frac{{{\sin{{2}}}{x}}}{{{\sin{{x}}}}}}={\frac{{{2}}}{{{\sec{{x}}}}}}?\)