# 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

S_{1}=cos(6)=0.96017

S_{2}=cos(6)+cos(12)

=0.9601+0.8439=1.80402

S_{3}=cos(6)+cos(12)+cos(18)

=1.804+0.6603=2.46434

S_{4}=cos(6)+cos(12)+cos(18)+cos(24)

=2.4643+0.4241=2.88852

S_{5}=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)

=2.888+0.1543=3.04277

S_{6}=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)

=3.0427-0.1280=2.91481

S_{7}=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)+cos(42)

=2.9147-0.3999=2.51483

S_{8}=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)+cos(42)+cos(48)

=2.5148-0.6401=1.87469

S_{9}=cos(6)+cos(12)+cos(18)+cos(24)+cos(30)+cos(36)+cos(42)+cos(48)+cos(54)

=1.8747-0.8293=1.04538

S_{10}=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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