Mathematical Series Shahbaz Ahmed Alvi
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What is a Series?
A mathematical series is basically a string of terms added together.
For example,
What is a Series?
The term on the right hand side of the equation is the general term of the summation. It is not always possible to have such a term.
Sometime the summations are infinite, like the ones we saw before.
Other times the series is finite, where it terminates at some nth term.
There are some common classes of series in Mathematics,
Common series
Arithmetic series,
Geometric series,
𝑛 𝑆𝑛 = 𝑎1 + 𝑎1 + 𝑑 + 𝑎1 + 2𝑑 + ⋯ + 𝑎1 + 𝑛 − 1 𝑑 = (2𝑎1 + 𝑛 − 1 𝑑 2 𝑆𝑛 = 𝑎 + 𝑎𝑟 +
𝑎𝑟 2
+
𝑎𝑟 3
+ ⋯+
𝑎𝑟 𝑛−1
1 − 𝑟𝑛 =𝑎 1−𝑟
There are several famous series used in Mathematics and Physics. For example,
Famous series
Taylor series, 𝑓′ 𝑎 𝑥 − 𝑎 𝑓 ′′ 𝑎 𝑥 − 𝑎 𝑓 𝑥 =𝑓 𝑎 + + 1! 2!
The black curve is sin 𝑥. Other curves are increasing approximation of 1, 3, 5, 7, 9, 11 and 13 terms of the corresponding Taylor series.
2
+⋯
Famous series
Fourier Series, 𝑎0 𝑆 𝑥 = + 2
∞
𝑎𝑛 cos 𝑛𝑥 + 𝑏𝑛 sin 𝑛𝑥 𝑛=1
Algebraic operations on a series
Adding or subtracting two series is done term by term. If there are two series, 𝑦 𝑥 = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + ⋯ = 𝑧 𝑥 = 𝑏0 + 𝑏1 𝑥 + 𝑏2 𝑥 2 + 𝑏3 𝑥 3 + ⋯ =
𝑎𝑚 𝑥 𝑚
𝑚=0
𝑏𝑚 𝑥 𝑚
𝑚=0
Then subtracting or adding the two series.
𝑦 𝑥 ± 𝑧 𝑥 = 𝑎0 ± 𝑏0 + 𝑎1 𝑥 ± 𝑏1 𝑥 + 𝑎2 𝑥 2 ± 𝑏2 𝑥 2 + ⋯ = 𝑦 𝑥 ±𝑧 𝑥 =
(𝑎𝑚 𝑥 𝑚 ± 𝑏𝑚 𝑥 𝑚 )
𝑚=0
(𝑎𝑚 ± 𝑏𝑚 )𝑥 𝑚
𝑚=0
If you multiply a constant number with a series, it is simply multiplied by every term in the series, 𝑏𝑦 𝑥 = 𝑏𝑎0 + 𝑏𝑎1 𝑥 + 𝑏𝑎2 𝑥 2 + ⋯ = 𝑏
𝑎𝑚 𝑥 𝑚 =
𝑚=0
𝑏𝑎𝑚 𝑥 𝑚
𝑚=0
Algebraic operations on a series
A series can also be multiplied by another series in which case each term of one series is multiplied by each term of the other series (notice two sets of indices), 𝑦 𝑥 𝑧 𝑥 =
𝑎𝑚 𝑥 𝑚
𝑚=0
𝑏𝑛 𝑥 𝑛 =
𝑛=0
𝑚=0
𝑏𝑛 𝑎𝑚 𝑥 𝑛 𝑥 𝑚
𝑛=0
There is a cleaner way of writing this, its called the Einstein’s convention. If two indices are repeated in an equation, it automatically means that they are being summed over. So, the series can simply be written as, 𝑦 𝑥 = 𝑎𝑚 𝑥 𝑚 and 𝑧 𝑥 = 𝑏𝑛 𝑥 𝑛
So the product can be written simply as, 𝑦 𝑥 𝑧 𝑥 = 𝑎𝑚 𝑏𝑛 𝑥 𝑚 𝑥 𝑛
And the sum is automatically implied. Makes it look cleaner if you have too many summation in your equation.
But Einstein summation convention is really only used in tensor analysis and General Relativity and we will rarely used it here in this course.
Algebraic operations on a series
Differentiation of a series is carried out by differentiating it term by term. The following series, 𝑦 𝑥 = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + ⋯ =
𝑚𝑎𝑚 𝑥 𝑚−1
𝑚=1
Similarly, second differential is, 𝑦 ′′
𝑚=0
Differentiates to, 𝑦′ 𝑥 = 0 + 𝑎1 + 2𝑎2 𝑥 + 3𝑎3 𝑥 2 + ⋯ =
𝑎𝑚 𝑥 𝑚
𝑥
= 0 + 2𝑎2 + 6𝑎3 𝑥 + ⋯ =
𝑚(𝑚 − 1)𝑎𝑚 𝑥 𝑚−2
𝑚=2
A series can also be integrated term by term, since the differential operator is linear. So, 𝑦 𝑥 𝑑𝑥 =
𝑎0 𝑑𝑥 +
𝑎1 𝑥 𝑑𝑥 +
𝑎2 𝑥 2 𝑑𝑥 + ⋯ =
𝑎𝑚 𝑥 𝑚 𝑑𝑥 =
𝑚=0
𝑚=0
𝑎𝑚
𝑥 𝑚 𝑑𝑥
Convergence and divergence of series
When talking about series, the single most important property of series is its convergence or divergence.
If the series approaches a certain number as more terms are added to it then the series is called convergent.
And if it doesn’t then the series is called divergent.
For example, consider the following series. 1 1 1 1 1 1+ + + + + … 2 4 8 16 32 The sum goes to infinity. But notice that, 1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + ⋯
Notice that although the series is infinite, increasing terms become progressively smaller.
Convergence and divergence of series 1.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + ⋯ 1.875 + 0.0625 + 0.03125 + ⋯ 1.9375 + 0.03125 + ⋯
Which means that later terms add less and less to the sum.
This means that even though the series is infinite, the sum will progressively approach some finite number and not infinity.
Such a series is called convergent because it “converges” to a point (a number).
As opposed to the previous example, consider the following simple series. 1+2+3+4+5+6+7+8+9+⋯
The series does not approach a finite number as you add more terms. Instead it approaches infinity.
This is a divergent series because the sum “diverges” as you add more terms to it.
Convergence and divergence of series
Series that are of importance to Physics and engineering should be convergent.
For example, the following function is the Taylor series of sin 𝑥. ∞ 𝑥 2𝑘+1 𝑘 sin 𝑥 = −1 2𝑘 + 1 ! 𝑘=0
This is how your calculators evaluate sine of angles.
There would be no point of this series if it did not converge to a number i.e. sine of an angle.
Test of convergence
So given a series, how do you determine if it will converge or not?
If you have an infinite series, then a series is said to be convergent if the sum of n terms approaches a finite number as n approaches infinity.
For example, consider a geometric series, 𝑎 + 𝑎𝑟 + 𝑎𝑟 2 + 𝑎𝑟 3 + ⋯
The sum of n terms is given as, 𝑆𝑛 = 𝑎 + 𝑎𝑟 +
𝑎𝑟 2
+
𝑎𝑟 3
1 − 𝑟𝑛 +⋯=𝑎 1−𝑟
If we take 𝑛 → ∞, then 𝑆𝑛 has 𝑟 𝑛 which will become infinitely large if 𝑟 > 1. Hence the sum will become infinitely large i.e. it will diverge.
But if 𝑟 < 1 then term becomes progressively small and sum approaches, 𝑎 𝑆𝑛→∞ = 1−𝑟
So the series is only convergent for some values of 𝑟 but not always!
Test of convergence: Comparison test
The simplest test of convergence is that of comparison.
Suppose that the following series is known to be convergent, 𝑎0 + 𝑎1 + 𝑎2 + ⋯ + 𝑎𝑛−1 + 𝑎𝑛 + ⋯
Suppose we need to determine the convergence of the following series, 𝑢0 + 𝑢1 + 𝑢2 + ⋯ + 𝑢𝑛−1 + 𝑢𝑛 + ⋯
The test then states that if a term by term comparison of the two series reveals that, 0 ≤ 𝑢𝑛 ≤ 𝑎𝑛
Then if
This also makes sense intuitively sense since every term is less than or equal to the “standard” series so if one converges so does the other.
𝑎𝑛 converges, so does the series
𝑢𝑛 . Otherwise it diverges.
Test of convergence: Comparison test
Suppose we want to determine if this series convergent, 1 1 1 1 + + + ⋯ = 12 22 32 𝑛2
For comparison we compare it with a series we know is convergent i.e. 1 𝑛(𝑛 + 1)
We see that, 𝑛(𝑛 + 1) ≥ 𝑛2
Which implies that, 1 1 ≤ 2 𝑛(𝑛 + 1) 𝑛
So every term in our series is either equal to or smaller than the “standard” 1 1 series. Hence we can conclude that if converges then so does 2 𝑛(𝑛+1)
𝑛
Test of convergence: Cauchy root test
Suppose there is a series such that, 0≤
1 𝑎𝑛 𝑛
≤𝑟 1 its divergent.
Test of convergence: Cauchy ratio test
Suppose that in a given series,
𝑟 is again some number which is less than one. This amounts to say that, 𝑎𝑛+1 < 𝑎𝑛
Which basically says that the terms become progressively small. This can be proven by comparing it with a geometric series because since in a geometric series, 𝑟 𝑛+1 =𝑟 𝑟𝑛
And we know that if 𝑟 < 1 then the geometric series is convergent. Hence by comparison we can say that 𝑎𝑛 is also convergent.
If the ratio is greater than one then the series diverges. And if it is zero, the test fails to determine the convergence of the series.
𝑎𝑛+1 0≤ ≤𝑟
