Series and Progressions
Today's post is about "Series". Hope you all loved my
first post on "Evolution of Mathematics".
I'm not going to talk about progressions in detail as the readers
would be well-equipped with knowledge of Arithmatic, Geometric and Harmonic
Progressions.
Series is infinite sequence of numbers(Complex) which either tends
to a number(also called as Convergent series) or tends to infinity(also called
as Divergent series). They are widely used in physics, computer science,
statistics and finance.
There are various tests available to prove whether a series is
converging or diverging like "nth term test" and "ratio-test".
According to "nth term test", If $\underset{n\to \infty
}{\mathop{\lim }}\,{{a}_{n}}\ne 0$ then the series diverges and according to
“ratio test”, If there exists a constant C <
1 such that $|{{a}_{n+1}}/{{a}_{n}}|<C$ for all sufficiently large n,
then $\sum{{{a}_{n}}}$ converges absolutely.
There
are various types of series that are there in Mathematics and still plenty of research is going on in
this area.
Some of them are:-
Γ Power Series
Γ Taylor Series
Γ Laurent Series
Γ Dirichlet Series
Γ Fourier Series
Power Series:-
A power series is a series of the
form
{\displaystyle
\sum _{n=0}^{\infty }a_{n}(x-c)^{n}.}
$\sum\limits_{n=0}^{\infty
}{({{a}_{n}}{{(x-c)}^{n}})}$
Taylor Series:- A function can be expressed as a series of polynomial which is
calculated from the function’s derivatives at each point. It is expressed as :-
$\sum\limits_{n=0}^{\infty
}{\frac{{{f}^{n}}(a)}{n!}}{{(x-a)}^{n}}$
When a=0, the above series becomes “Maclaurin series”.
Laurent Series:-
This is a generalization
of Power Series that it accounts for both positive and negative terms. It is of
the form:-
$\sum\limits_{n=-\infty
}^{\infty }{{{a}_{n}}{{x}^{n}}}$
Dirichlet Series:- It is of the form
$\sum\limits_{n=1}^{\infty
}{\frac{{{a}_{n}}}{{{n}^{s}}}}$
Where ‘s’ is any complex number.
If all the an are equal to 1, then the Dirichlet Series becomes Riemann Zeta Function((ΞΆ(s)))
$\text{
}\!\!\zeta\!\!\text{ }\left( \text{s} \right)=\sum\limits_{n=1}^{\infty
}{\frac{1}{{{n}^{s}}}}$
This function is one of the most trending functions in this
century. The solution of this function is still a mystery and it is one of the
unsolved problems in Mathematics today.
More on this function will be discussed in future posts.
Fourier Series:-
It is a way of expressing
periodic functions as an infinite series of sum of sine and cosine functions with corresponding
co-efficients. It is of the form:-
$f(x)={{A}_{0}}+\sum\limits_{n=1}^{\infty
}{({{A}_{n}}\cos nx+{{B}_{n}}\sin nx)}$
More on Fourier series will be discussed in my next post.
So far the series we have discussed are infinite summation series.
But there are other types of expressions also like infinite product series,
infinite nested radicals, infinite continued fractions, infinite power towers
etc.
Infinite Product Series:-
It is of the form:-
$\prod\limits_{n=1}^{\infty
}{{{a}_{n}}={{a}_{1}}.{{a}_{2}}.{{a}_{3}}...}$
The infinite product is said to converge if and only if the sum
$\sum\limits_{n=1}^{\infty
}{\ln ({{a}_{n}})}$
Converges.
Infinite Nested Radicals:-
Ramanujan's infinite radicals
Ramanujan posed the following problem
to the Journal of Indian Mathematical Society:
$?=\sqrt{1+2\sqrt{1+3\sqrt{1+....}}}$
{\displaystyle ?={\sqrt {1+2{\sqrt {1+3{\sqrt
{1+\cdots }}}}}}.\,}
This can be solved by noting a more
general formulation:
{\displaystyle
?={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\mathrm {\cdots }
}}}}}}.\,}?$?=\sqrt{ax+{{(n+a)}^{2}}+x\sqrt{a(x+n)+{{(n+a)}^{2}}+(x+n)\sqrt{.....}}}$
Setting this to F(x)
and squaring both sides gives us
{\displaystyle
F(x)^{2}=ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\mathrm {\cdots }
}}}},\,} $F{{(x)}^{2}}=ax+{{(n+a)}^{2}}+x\sqrt{a(x+n)+{{(n+a)}^{2}}+(x+n)\sqrt{.....}}$
which can be simplified to
$F{{(x)}^{2}}=ax+{{(n+a)}^{2}}+xF(x+n)$
{\displaystyle
F(x)^{2}=ax+(n+a)^{2}+xF(x+n).\,}xxxxx$F{{(x)}^{2}}=ax+{{(n+a)}^{2}}+xF(x+n)$
It can then be shown that
{\displaystyle
F(x)={x+n+a}.\,}F(x) = x + n + a
So, setting a = 0, n = 1,
and x = 2, we have
{\displaystyle
3={\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}.\,}
$3=\sqrt{1+2\sqrt{1+3\sqrt{1+.....}}}$
Continued Fractions will be
discussed exclusively in future posts.
Hope you have enjoyed the introduction of series in this post.
I’ll be back with problems as well as in depth discussion of some
of the series discussed above.
Please comment below if you have any queries.
Watch these two videos for some amazing facts about Series:-
