Mathematical Formulas for Industrial and Mechanical Engineering

5.25 Some of Finite and Infinite Series

1 1 2 1 3 1 ? 1 n 5

nðn 1 1Þ

2

2 1 4 1 6 1 ? 1 2n 5 nðn 1 1Þ

1 1 3 1 5 1 ? 1 ð2n 2 1Þ5 n

2

k 1 ðk 1 1Þ1 ðk 1 2Þ1 ? 1 ðk 1 n 2 1Þ5

nð2k 1 n 2 1Þ

2

1

2

1 2

2

1 3

2

1 ? 1 n

2

5

nðn 1 1Þð2n 1 1Þ

6

1

3

1 2

3

1 3

3

1 ? 1 n

3

5

nðn11Þ

2



2

1

2

1 3

2

1 5

2

1 ? 1 ð2n21Þ

2

5

nð4n

2

2 1Þ

3

1

3

1 3

3

1 5

3

1 ? 1 ð2n21Þ

3

5 n

2

ð2n

2

2 1Þ

1 1

1

2

1

1

2

2

1

1

2

3

1 ? 1

1

2

n

1 ? 5 2

1 1

1

1!

1

1

2!

1

1

2!

1 ? 1

1

ðn 2 1Þ!

1 ? 5 e

5.26 Convergence Tests for Series

A series converges iff the associated sequence of partial sums represented by {S

k

}

converges. The element S

k

in the sequence above is defined as the sum of the first

“k” terms of the series.

5.27 Series Tests

In this section the various tests mentio ...