The answer is listed. I need assistance with the process. a,) (15) (a) Consider

Question

The answer is listed. I need assistance with the process.

a,) (15) (a) Consider the series ?(-1). number of terms needed to approximate the sum of the series with an error of magnitude less than or equal to 3x10-7? Also give the value of that partial sum, rounded to 3 decimal places. (b) Repeat part (a) for the series yl, and decide the minimum number of terms needed to Use your calculator to decide the minimum (2n)! approximate the sum of the series with an error of magnitude less than or equal to 3x 10s

Explanation / Answer

In numerical methods, there is alwasy a discrepency between true value and estimated/approximated value. These discrepency refered as errors between true and estimated value.

In this example, error is widely known as truncation error( is the error made by truncating an infinite sum and approximating it by a finite sum).

Steps: This is an alterating series.

S= a_0 - a_1+a_2-a_3+a_4 - a_5+a_6-a_7 + ............

Now we have to check the values of a_n, n=0,1,2,,. If the absolute value of a_i is less than 3*10^{-7) (given that this condition). Then our job is done and just add up the values upto a_0 to a_(i-1}. This will give the approximated value of S and the truncation error is a_i.

Here, a_6 = -2.8 * 10^{-7} < 3 * 10^ {-7}.

Then approximated value of S= a_0 - a_1+a_2 -a_3+a_4-a_5. The truncation errors is a_6. ( approximated value is correct upto six decimal as six zeros is present after decimal).

Remark: If no condition is given and we have to find the approximated value is correct upto some decimal places.

If we have to find the approximated value is correct upto five decimal places.

In this case we calculate a_0, a_1,........

If a_9 = 0.0000167 and a_10= 0.00000156 will come ( no need to check the -ve or +ve sign, just check the absolute value)

Here, The term a_10 have five zero after decimal, it means that now we can add a_0 to a_9 and this will give the approximated value correct upto five decimal places. ( this method also sometimes called as first neglected term)

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