Find 10 partial sums of the series. (Round your answers to five decimal places.) \(\sum_{n=1}^{\infty }cos(6n)\). Also, graph both the sequence of terms and the sequence of partial sums.
Solution:
Given, \(\sum_{n=1}^{\infty }cos(6n)\)
We have to find the 10 partial sums of the series and graph the sequence of terms and the sequence of partial sums.
When n = 1, cos(6n) = cos(6) = 0.96017
When n = 2, cos(6n) = cos(12) = 0.84385
When n = 3, cos(6n) = cos(18) = 0.66032
When n = 4, cos(6n) = cos(24) = 0.42418
When n = 5, cos(6n) = cos(30) = 0.15425
When n = 6, cos(6n) = cos(36) = -0.12796
When n = 7, cos(6n) = cos(42) = -0.39998
When n = 8, cos(6n) = cos(48) = -0.64014
When n = 9, cos(6n) = cos(54) = -0.82931
When n = 10, cos(6n) = cos(60) = -0.95241
The first 10 partial sums are
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
Therefore, the first 10 partial sums are 0.96017, 1.80402, 2.46434, 2.88852, 3.04277, 2.91481, 2.51483, 1.87469, 1.04538 and 0.09297
Find 10 partial sums of the series. (Round your answers to five decimal places.) \(\sum_{n=1}^{\infty }cos(6n)\). Also graph both the sequence of terms and the sequence of partial
Summary:
The first 10 partial sums of the series. (Round your answers to five decimal places.) \(\sum_{n=1}^{\infty }cos(6n)\) are 0.96017, 1.80402, 2.46434, 2.88852, 3.04277, 2.91481, 2.51483, 1.87469, 1.04538 and 0.09297
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