First Proof (C. F. Gauß) for :
General proof by induction:
1. Induction Start ( ):
2. Induction Step ( ): Assume Eq. ( 4.1 ) is true for , then show that it is also true for :
This can now be used to prove it is true for , etc.
Proof: Write
Alternative proof by induction: home exercise.
Here, we define the binomial coefficient
The proof of Eq.( 4.7 ) goes again via induction . Not shown here. Examples:
A series
is called infinite series. It is the limit of the sequence of finite series when the upper limit tends toward infinity. (The objects are also called “partial sums”.) In contrast to the finite series , the infinite series can diverge. is said to be convergent is approaches a finite limit as .
Example 4.1:
Use a constant
is divergent because the partial sums , which clearly diverge as .
Example 4.2:
The geometric series
learn this one by heart.
This series converges for arbitrary
(real or complex) numbers
with
. [Please sketch the condition
for complex
as an an area in the complex plane.]
Proof of Eq. ( 4.33 ):
The problem with infinite series is that often it is not easy to decide if or if not they converge, e.g. for which values of in the above example.
A necessary condition for convergence of is that as .
A sufficient condition for convergence: is the ratio test
Ratio test: Consider the series and assume for all . Define the ratio
For , the ratio test can’t decide whether the series is convergent or divergent.