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Calculus hw, need help asap with steps.

Calculus hw, need help asap with steps.-example-1
User Null Canvas
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8 votes

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S1 = 1

S2 = 0.5

S3 = 0.6667

S4 = 0.625

S5 = 0.6333

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Step-by-step explanation:

Let
f(n) = ((-1)^(n+1))/(n!)

The summation given to us represents the following


\displaystyle \sum_(n=1)^(\infty) ((-1)^(n+1))/(n!)=\sum_(n=1)^(\infty) f(n)\\\\\\\displaystyle \sum_(n=1)^(\infty) ((-1)^(n+1))/(n!)=f(1) + f(2)+f(3)+\ldots\\\\

There are infinitely many terms to be added.

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The partial sums only care about adding a finite amount of terms.

The partial sum
S_1 is the sum of the first term and nothing else. Technically it's not really a sum because it doesn't have any other thing to add to. So we simply say
S_1 = f(1) = 1

I'm skipping the steps to compute f(1) since you already have done so.

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The second partial sum is when things get a bit more interesting.

We add the first two terms.


S_2 = f(1)+f(2)\\\\S_2 = 1+(-(1)/(2))\\\\S_2 = (1)/(2)\\\\S_2 = 0.5\\\\\\

The scratch work for computing f(2) is shown in the diagram below.

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We do the same type of steps for the third partial sum.


S_3 = f(1)+f(2)+f(3)\\\\S_3 = 1+(-(1)/(2))+(1)/(6)\\\\S_3 = (2)/(3)\\\\S_3 \approx 0.6667\\\\\\

The scratch work for computing f(3) is shown in the diagram below.

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Now add the first four terms to get the fourth partial sum.


S_4 = f(1)+f(2)+f(3)+f(4)\\\\S_4 = 1+(-(1)/(2))+(1)/(6)-(1)/(24)\\\\S_4 = (5)/(8)\\\\S_4 \approx 0.625\\\\\\

As before, the scratch work for f(4) is shown below.

I'm sure you can notice by now, but the partial sums are recursive. Each new partial sum builds upon what is already added up so far.

This means something like
S_3 = S_2 + f(3) and
S_4 = S_3 + f(4)

In general,
S_(n+1) = S_(n) + f(n+1) so you don't have to add up all the first n terms. Simply add the last term to the previous partial sum.

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Let's use that recursive trick to find
S_5


S_5 = [f(1)+f(2)+f(3)+f(4)]+f(5)\\\\S_5 = S_4 + f(5)\\\\S_5 = (5)/(8) + (1)/(120)\\\\S_5 = (19)/(30)\\\\S_5 \approx 0.6333

The scratch work for f(5) is shown below.

Calculus hw, need help asap with steps.-example-1
User StefansArya
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