Prove that $\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad-1<x<1$.

Find det(A), where:

$$A=\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 2 \\ 2 & 1 & 1 \end{array}\right]$$

Prove that if $| z | = 1 ( z \neq 1 ) , \text { then } \operatorname { Re } [ 1 / ( 1 - z ) ] = 1 / 2$.

Show that $\frac{n^{m+1}}{m+1}\lt 1^m+2^m+...n^m\lt \frac{(n+1)^{m+1}}{m+1}$ where $m$ is a positive integer.

Verify the conjecture that $1^{n-1}+2^{n-1}+3^{n-1}+\cdots+(n-1)^{(n-1)} \not \equiv-1(\bmod n)$ if n is composite, for as many integers n as you can.

Simplify the expression.

$$\left[ \left( x ^ { - 1 } + 1 \right) ^ { - 1 } + 1 \right] ^ { - 1 }$$

Consider the geometric series -1 + 1 – 1 + 1 – 1 + . . . Describe any conclusions.

Simplify the expression. $\frac{1}{1+\frac{1}{1+\frac{1}{3}}}$

Prove:

$$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) , \quad - 1 < x < 1$$

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Use the formula

$$\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \quad-1<x<1$$

and the Maclaurin series for $\ln$(1 + x) to show that

$$\tanh ^{-1} x=\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{2 n+1}$$

Simplify.

$$\frac{1-\frac{1}{y}+\frac{1}{y^2}-\frac{1}{y^3}}{1-\frac{1}{y^4}}$$

If $x \sqrt{1+y}+y \sqrt{1+x}=0$, for, $-1<x<1$, prove that

$$\frac{d y}{d x}=-\frac{1}{(1+x)^2}$$

Determine whether the given points are coplanar. A(0, 1, 2), B(-1, 1, 0), C(2, 0, -1), D(1, -1, 1)

Prove that $\tanh ^{-1} x=\dfrac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad-1<x<1$.