Question

Estimate the sum from n=1 to infinity of (2n+1)^(−9) correct to five decimal places.

Answer 1

\[\sum_{n=1}^{\infty}(2n+1)^{-9}\]

Answer 2

The remainder (R) of a series that converges by the integral test of the first n terms can be estimated as follows:
$$
\Large \int_{n+1}^{\infty}f(x) dx \leq R \leq \int_{n}^{\infty} f(x) dx
$$

Answer 3

Since we require 5 digit accuracy, we should aim for R < 0.00001. I'll use 0.000005.

Answer 4

$$
\Large{ \int_{n+1}^{\infty}f(x) dx \leq R_n \leq \int_{n}^{\infty} \frac{1}{(2x+1)^9} dx
\leq 0.000005
\\~\\ \therefore \\ ~\\
\left[ \frac{-1}{(16(2x+1)^8)} \right]_{n}^{\infty}\leq 0.000005
\\ ~\\ \therefore

\frac{-1}{(16(2 \cdot \infty +1)^8)} - \frac{-1}{(16(2n+1)^8)} \leq 0.000005
\\ ~\\ \therefore \\
0 + \frac{1}{(16(2n+1)^8)} \leq 0.000005

\\ ~\\ \therefore \\
\frac{1}{(16(2n+1)^8)} \leq 0.000005


\\ ~\\ \therefore \\
\frac{1}{(16(2n+1)^8)} \leq \frac{5}{1000000}


}

$$

Answer 5

ok i got\[\frac{ 1 }{ (2n+1)^{8} }\le \frac{ 1 }{ 12500 }\]
\[(2n+1)^{8}\ge12500\]
\[2n+1\ge3.251725\]
\[2n \ge2.251725\]
\[n \ge1.125862\]

Answer 6

can someone please help me with what to do next

Answer 7

If all of that is correct, then you want to sum up the terms from n=1 to n=2

Answer 8

okay so i use \[\frac{ 1 }{ (2(1)+1)^{8} }+\frac{ 1 }{ (2(2)+1)^{8} }\]

Answer 9

I think the original series has 9 not 8 as the exponent

Answer 10

oh yeah so that would equal 0.0000513

Answer 11

?

Answer 12

do i do anything after that?

Answer 13

that looks about right
if you add the next term you will notice the 5 does not change

Answer 14

but when i typed this in the answer was wrong

Answer 15

someone please help?

Answer 16

what is the value when n=1?

Answer 17

1/19683

Answer 18

yeah thats not in decimal form is it ....

Answer 19

0.0000508

Answer 20

now compare this to what you got when n=2, do you see that they are the same for the first 5 decies?

Answer 21

yes

Answer 22

then n=2 is not the answer we are looking for

Answer 23

but when i typed this in it said the answer was wrong

Answer 24

0.00005 is correct to 5 decimal places

Answer 25

im not sure how your program is wanting the answer formated. but im pretty sure that the estimation is: .00005

Answer 26

can you screenshot it?

Answer 27

yeah, you want it accurate to 5 decimal places, you arent putting in exactly 5 decimal places are you?

Answer 28

0.12345
0.00005

Answer 29

i do you need help still