Find at least 10 partial sums of the series. (round your answers to five decimal places.)

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30-03-2022
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Find at least 10 partial sums of the series. (round your answers to five decimal places.)

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Martebi Martebi · Answer 1 · 2022-04-04T12:12:01+02:00

Since the equation is not given. To find the 10 partial sums of the series when ∑∞ₙ= ₁ cos(6) is given below:

The 10 partial sums to be 0.96017, 1.80402, 2.46434, 2.88852, 3.04277, 2.91481, 2.51483, 1.87469, 1.04538 and 0.09297.

What is a Partial Sum?

This is known to be the sum (addition) of a section of the sequence. The addition of infinite terms is known to be an Infinite Series. Partial Sums are often known as Finite Series.

Note that n= 1,2,3,4...10, so we would fit the numbers into the equation given above:

If n = 1, cos(6n) = cos(6) = 0.96017

if n = 2, cos(6n) = cos(12) = 0.84385

if n = 3, cos(6n) = cos(18) = 0.66032

if n = 4, cos(6n) = cos(24) = 0.42418

if n = 5, cos(6n) = cos(30) = 0.15425

if n = 6, cos(6n) = cos(36) = -0.12796

if n = 7, cos(6n) = cos(42) = -0.39998

if n = 8, cos(6n) = cos(48) = -0.64014

if n = 9, cos(6n) = cos(54) = -0.82931

if n = 10, cos(6n) = cos(60) = -0.95241

Since we know what the values of N are, we can then solve for the first 10 partial sums using: We take partial sum to be (S). So therefore:

S1=cos(6)=0.96017

S2=cos(6)+cos(12) =0.9601+0.8439=1.80402

S3=cos(6)+cos(12)+cos(18) =1.804+0.6603=2.46434

S4=cos(6)+cos(12)+cos(18)+cos(24) =2.4643+0.4241=2.88852

S5=cos(6)+cos(12)+cos(18)+cos(24)+cos(30) =2.888+0.1543=3.04277

S6=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)=3.0427-0.1280=2.91481

S7=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)+cos(42)=2.9147-0.3999=2.51483

S8=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)+cos(42)+cos(48) =2.5148-0.6401=1.87469

S9=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)+cos(42)+cos(48)+cos(54) =1.8747-0.8293=1.04538

S10=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)+cos(42)+cos(48)+cos(54)+cos(60) =1.0454-0.9524=0.09297

So therefore, we have the 10 partial sums to be 0.96017, 1.80402, 2.46434, 2.88852, 3.04277, 2.91481, 2.51483, 1.87469, 1.04538 and 0.09297 respectively.

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