Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.4 OPERATIONS WITH SERIES is always less thanuNfor allmandun→0asn→∞, the alternating series converges. It is clear that an anal ...

SERIES AND LIMITS although in principle infinitely long, in practice may be simplified ifxhappens to have a value small compared ...

4.5 POWER SERIES
Determine the range of values ofxfor which the following power series converges: P(x)=1+2x+4x^2 +8x^3 +···. By ...

SERIES AND LIMITS r=−expiθ. Therefore, on the the circle of convergence we have P(z)= 1 1+expiθ . Unlessθ=πthis is a finite comp ...

4.5 POWER SERIES Q(x)intoP(x)toobtainP(Q(x)), but we must be careful since the value ofQ(x) may lie outside the region of conver ...

SERIES AND LIMITS 4.6 Taylor series Taylor’s theorem provides a way of expressing a function as a power series inx, known as aTa ...

4.6 TAYLOR SERIES f(a) f(x) a a+h x P Q R θ hf′(a) h Figure 4.1 The first-order Taylor series approximation to a functionf(x). T ...

SERIES AND LIMITS x=a+hin the above expression. It then reads f(x)=f(a)+(x−a)f′(a)+ (x−a)^2 2! f′′(a)+···+ (x−a)n−^1 (n−1)! f(n− ...

4.6 TAYLOR SERIES We may follow a similar procedure to obtain a Taylor series about an arbitrary pointx=a.
Expandf(x)=cosxas a ...

SERIES AND LIMITS value ofξthat satisfies the expression forRn(x) is not known, an upper limit on the error may be found by diff ...

4.7 EVALUATION OF LIMITS These can all be derived by straightforward application of Taylor’s theorem to the expansion of a funct ...

SERIES AND LIMITS
Evaluate the limits lim x→ 1 (x^2 +2x^3 ), lim x→ 0 (xcosx), lim x→π/ 2 sinx x . Using (a) above, lim x→ 1 (x ...

4.7 EVALUATION OF LIMITS Therefore we find lim x→a f(x) g(x) = f′(a) g′(a) , providedf′(a)andg′(a) are not themselves both equal ...

SERIES AND LIMITS Summary of methods for evaluating limits To find the limit of a continuous functionf(x) at a pointx=a, simply ...

4.8 EXERCISES 4.8 TheN+ 1 complex numbersωmare given byωm=exp(2πim/N), form= 0 , 1 , 2 ,... ,N. (a) Evaluate the following: (i) ...

SERIES AND LIMITS 4.15 Prove that ∑∞ n=2 ln [ nr+(−1)n nr ] is absolutely convergent forr= 2, but only conditionally convergent ...

4.8 EXERCISES 4.20 Identify the series ∑∞ n=1 (−1)n+1x^2 n (2n−1)! , and then, by integration and differentiation, deduce the va ...

SERIES AND LIMITS (a) lim x→ 0 sin 3x sinhx , (b) lim x→ 0 tanx−tanhx sinhx−x , (c) lim x→ 0 tanx−x cosx− 1 , (d) lim x→ 0 ( cos ...

4.9 HINTS AND ANSWERS find a closed-form expression forα, the Madelung constant for this (unrealistic) lattice. 4.35 One of the ...

SERIES AND LIMITS 4.15 Divide the series into two series,nodd andneven. Forr= 2 both are absolutely convergent, by comparison wi ...