Some basics on infinite series (not covered in the textbook)

Only in rare occasions can we sum up an infinite series in a closed form (Fourier series, for example) hence it is necessary to carry out numerical approximation for most of infinite series. Therefore, it is extremely important to be able to know in advance whether a given infinite series converges or diverges.
Example
100
∑
i=1 
 1
i
 ≈ 5.187,     1000
∑
i=1 
 1
i
 ≈ 7.485,     5000
∑
i=1 
 1
i
 ≈ 9.094     10000
∑
i=1 
 1
i
 ≈ 9.787,
From the above, one tends to think that ∑1/n converges to a value around 10. In fact, the series ∑1/n diverges as n → ∞ but its divergence speed is so slow that an illusion follows.

The Riemann zeta (ζ) function (p-series)

The most important infinite series that can be used as a reference series is the Riemann zeta (ζ) function defined as
ζ(p) ≡ ∞
∑
n=1 
 1
np
 .
(1)
Theorem: (Important) The p-series converges when p > 1.
(Proof) pseries1.gif
More examples
∑
(n + 3)−3/2
∑
 n
n2 + 3 ln n
∑
n−100
∑
(1 + 1
n2
 )
∑
⎛
⎝ 		1
n
 − 1
n + 10
⎞
⎠
∑
⎛
⎝ 		cos n
2 n − 1
⎞
⎠ 		2

 
∑
 (−1)n+1
n
∑
 n + cos2  n
n2 + 4
∑
e−n x
∑
ln (2 + 2
n
 )


File translated from TEX by TTH, version 4.03.
On 21 Aug 2022, 22:51.