Mathematics, 08.10.2019 14:20 exoticbunnylover123

Estimate the value of the following convergent series with an absolute error less than 10^-4

∞
∑ )^k)/(2k+1)!
k=1

i set my ak=1/(2k+ then sub in n+1 for k because i am approximating the value to 10^-4. however, i don't know how to handle the factorial. i am just left with (2n+3)! =10^4 and don't know how to solve for just

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Answer from: kaylesparks2

We can find the number of terms needed to fall within that accuracy. Taking the absolute difference of the

th and

th partial sums, you have

$\displaystyle\left|\sum_{k=1}^{n+1}\frac{(-1)^k}{(2k+1)!}-\sum_{k=1}^n\frac{(-1)^k}{(2k+1)!}\right|=\frac1{(2n+3)!}$

or equivalently,

(2n+3)!10000

It's easy enough to check that (2n+2)!

exceeds 10000 for all $n\ge3$ , since $(2\times3+2)!=8!=362880$ . This means the sum can be approximated to within $10^{-4}$ with just the first three terms.

You have

$\displaystyle\sum_{k=1}^3\frac{(-1)^k}{(2k+1)!}=-\frac1{3!}+\frac1{5!}-\frac1{7!}=-\frac1{7!}\left(\frac{7!}{3!}-\frac{7!}{5!}+1\right)=-\frac{799}{5040}\approx-0.158532$

The actual value of the sum is $\sin(1)-1$ , which comes from the fact that

$\sin x=\displaystyle\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!}\implies \sin x-x=\sum_{k=1}^\infty\frac{(-1)^kx^{2k+1}}{(2k+1)!}$

and plugging in x=1

. The actual value is approximately -0.158529

which verifies that the approximation is within 4 decimal places of the infinite series.

Answer from: Quest

the required function is $y=-x+4$ .

step-by-step explanation:

from the given table it is noticed that the graph passing through (-3,7), (0,4), (3,1) and (8,-4).

it is a linear function because the value of y is decrease by one as the value of x increase by 1.

the slope of the function is

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-7}{)}=\frac{-3}{3}=-1$

the slope of the function is -1 and the y-intercept is 4.

the slope intercept form is

$y=mx+b$

where, m is slope and b is y-intercept.

$y=-1x+4$

$y=-x+4$

therefore the required function is $y=-x+4$ . the graph of this function is shown below.

Answer from: Quest

see below

step-by-step explanation:

7. there are 10 data points in the set. the middle point is between 5 and 6. the middle point in the top half is 8 and the middle point in the bottom half is 3. to find the iqr , take point 8 and subtract point 3

mid term point 8 = 100 point 3 = 88

iqr = 100-88 = 12

final point 8 = 93 point 3 = 78

iqr = 93 - 78 = 15

the final has the greatest iqr

6. b - there is a high data value that causes the data set to be asymmetrical for the males: the data for males are high and asymmetrical . there is an outlier for the males that skews the data

5. the iqr is a better measure of spread for the movies and basketball games. since the basketball game has an outlier, we do not want to use standard deviation for measurement.

3. the exam median is much higher than the class median. the median line is farther to the right on the exams

2. there is no way to tell what the means are from the box and whisker plots.