Ch4-Series and limits

Mathematical Methods for Physics and Engineering

1. Basic sum of series [Page 117-118]:

Basic series:

Especially, S_N=\frac{a}{1-r} for infinite geometric series with|r|<1.

Arithmetico-geometric series:

Especially, S_N=\frac{a}{1-r}+\frac{rd}{(1-r)^2} for infinite arithmetico-geometric series with |r|<1.

2. Some useful methods [Page 119-124]:

2.1 The difference method

For series u_n, if you can find u_n=f(n)-f(n-1), then \sum\limits_{n=1}^{N}u_n=f(N)-f(0).

(Try it \sum\limits_{n=1}^{N}\frac{1}{n(n+1)(n+2)}, \sum\limits_{n=1}^{N}n^3 )

Example:

I will show you how to find \sum\limits_{n=1}^{N}n^3 which isn't stated clearly on page 121.

Similarly,

2.2 The transformation method

This is not only limited to few ways. Here's three examples.

Example:

1. Find S(x)=\frac{x^4}{3(0!)}+\frac{x^5}{4(1!)}+\frac{x^6}{5(2!)}+....

With \frac{S(x)}{x}=0 at x=0, we finally find S(x)=x^3e^x-2x^2e^x+2xe^x-2x.

2. Find S=1+\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+....

You can check the result by the formula in section 1: S=\frac{a}{1-r}+\frac{rd}{(1-r)^2}=\frac{1}{1-\frac{1}{2}}+\frac{\frac{1}{2}\times1}{(1-\frac{1}{2})^2}=4.

3. Find S(\theta)=1+cos\theta+\frac{cos2\theta}{2!}+\frac{cos3\theta}{3!}+....

3. Convergence of infinite series [Page 124-131]:

3.1 The definition:

Absolute convergence: If the series \sum|u_n| converges then \sum u_n also converges, and \sum u_n is said to be absolutely convergent.

Conditional convergence: if \sum |u_n|  diverges whilst \sum u_n converges then \sum u_n is said to be conditionally convergent.

3.2 Test of convergence:

3.2.1 Preliminary test:

For positive terms only series (necessary but not sufficient): \sum u_n is convergent if \lim\limits_{n\rightarrow\infty}u_n=0.

3.2.2 Comparison test:

For positive terms only series: \sum u_n is convergent if \sum v_n is convergent and u_n < v_n(\forall n>N,N is finite).

3.2.3 D’Alembert’s ratio test:

For positive terms only series: define \rho=\lim\limits_{n\rightarrow\infty}\frac{u_{n+1}}{u_n},

(i) if \rho<1, then u_n is convergent.

(ii) if \rho>1, then u_n is divergent.

(iii) if \rho=1, then u_n is undetermined.

3.2.4 Ratio comparison test:

For positive terms only series: \sum u_n is convergent if \sum v_n is convergent and \frac{u_{n+1}}{u_n} \leq \frac{v_{n+1}}{v_n}(\forall n>N,N is finite); \sum u_n is divergent if \sum v_n is dinvergent and \frac{u_{n+1}}{u_n} \geq \frac{v_{n+1}}{v_n}(\forall n>N,N is finite).

3.2.5 Quotient test:

For positive terms only series: define \rho=\lim\limits_{n\rightarrow \infty}\frac{u_n}{v_n},

(i) if \rho \neq 0 but is finite then \sum u_n and \sum v_n both converge or both diverge.

(ii) if \rho=0 and v_n converges then u_n converges.

(iii) if \rho=\infty and v_n diverges then u_n diverges.

3.2.6 Integral test:

For positive terms only series: \sum u_n is convergent if u_n = f(n) and \lim\limits_{N\rightarrow \infty}\int^{N}f(x)dx exists.

Example:

Determine if \sum\limits_{n=1}^{\infty}\frac{1}{n^p} (Riemann zeta series) is convergent.

which means \sum\limits_{n=1}^{\infty}\frac{1}{n^p} is convergent for p>1 and divergent for p\leq1.

3.2.7 Cauchy’s root test:

For positive terms only series: define \rho=\lim\limits_{n\rightarrow \infty}(u_n)^{1/n},

(i) if \rho<1, then u_n is convergent.

(ii) if \rho>1, then u_n is divergent.

(iii) if \rho=1, then u_n is undetermined.

3.2.8 Grouping items:

Let's get back to the Riemann zeta series stated before. Since

which converges for p>1, then \frac{1}{n^p} must also converges for p>1. Since

which diverges, then \frac{1}{n^p} must also diverges for p\leq1.

3.2.9 Alternating series test:

\sum\limits_{n=1}^{\infty}(-1)^{n+1}u_n can be shown to converge provided:

(i) \lim\limits_{n\rightarrow\infty}u_n=0.

(ii) u_n < u_{n−1} for all n > N for some finite N.

See page 131 for more details.

3.2.10 Power series test:

If P(z)=a_0+a_1z+a_2z^2+\cdots where z is a complex number, then P(z) converges for:

where \rho is called radius of convergence.

Example:

Determine the range of z for which the following complex power series converges:

Answer:

It's easy to know it converges for |z|<2. The problem is how to investigate the condition on the circle |z|=2. We can write z=2e^{i\theta}, then:

We find it converges at all points on the circle except \theta=\pi which makes P(z) infinite.

4. Cauchy product [Page 131]:

If \sum u_n converges to S and \sum v_n converges to T, then the Cauchy product:

converges to ST.

5. Operations with power series [Page 134-135]:

If two power series P(x) and Q(x) are convergent, then the sum, difference, substitution, differentiation and integration also converge to the same range where P(x)=a_0+a_1x+a_2x^2+\cdots, so as Q(x).

6. Taylor series [Page 138-140]:

where R_n(x)=\frac{(x-a)^{n}}{n!}f^{(n)}(\xi),\ \xi \in[a,x] called Lagrange remainder, or R_n(x)=o(x^{n-1}),\ \xi \in[a,x] called Peano remainder.

If set a = 0, then we obtain Maclaurin series:

with error estimation |R_n(x)|\leq \frac{M}{n!}x^n.

Here's some important Maclaurin formulas:

7. Evaluation of limits [Page 142]:

7.1 Use ln: \lim\limits_{x\rightarrow \infty}y(x)=\lim\limits_{x\rightarrow \infty}e^{ln[y(x)]}.

7.2 Use l'Hopital's rule: \lim\limits_{x\rightarrow \infty}\frac{f(x)}{g(x)}=\lim\limits_{x\rightarrow \infty}\frac{f'(x)}{g'(x)}.