Raul Walker

2022-07-08

Calculate the difference between π/4 and the Leibniz series for computing $\pi /4$ with $n=200$.

This series appears to converge relatively slowly, and so at what point can we confidently say that "the $576$th digit is $3$" of an irrational number?

This series appears to converge relatively slowly, and so at what point can we confidently say that "the $576$th digit is $3$" of an irrational number?

postojahob

Beginner2022-07-09Added 13 answers

The way you can tell how many digits you have computed is by providing a bound on the remainder term for the series (i.e., of ${R}_{k}=\sum _{n=k+1}^{\mathrm{\infty}}{a}_{n}$). Say you want m digits, then you want ${10}^{-m}>\left|{R}_{k}\right|$.

For the Liebniz series $\sum _{n=0}^{\mathrm{\infty}}\frac{(-1{)}^{n}}{2n+1}$, since it is an alternating series, we know that the remainder term is bounded by $\frac{1}{2n+1}$ itself, and so if you want $m$ digits of $\frac{\pi}{4}$, $\frac{{10}^{m}-1}{2}$ terms would be sufficient (quite a few).

For other series, sometimes Tayor's theorem can be used to provide a bound on the remainder term.

For the Liebniz series $\sum _{n=0}^{\mathrm{\infty}}\frac{(-1{)}^{n}}{2n+1}$, since it is an alternating series, we know that the remainder term is bounded by $\frac{1}{2n+1}$ itself, and so if you want $m$ digits of $\frac{\pi}{4}$, $\frac{{10}^{m}-1}{2}$ terms would be sufficient (quite a few).

For other series, sometimes Tayor's theorem can be used to provide a bound on the remainder term.

Find the volume V of the described solid S

A cap of a sphere with radius r and height h.

V=??

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$In how many different orders can five runners finish a race if no ties are allowed???

State which of the following are linear functions?

a.$f(x)=3$

b.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

d.$t(x)=5(x-2)$ Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?

A square is also a

A)Rhombus;

B)Parallelogram;

C)Kite;

D)none of theseWhat is the order of the numbers from least to greatest.

$A=1.5\times {10}^{3}$,

$B=1.4\times {10}^{-1}$,

$C=2\times {10}^{3}$,

$D=1.4\times {10}^{-2}$Write the numerical value of $1.75\times {10}^{-3}$

Solve for y. 2y - 3 = 9

A)5;

B)4;

C)6;

D)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.