Notice that the sequence (an)∞n=0may be written as (a0, a1, a2

(1) Let V be the space of all sequences (an)^∞_n=0of real numbers. On V, we define (a_n)^∞_n=0+ (b_n)^∞_n=0= (a_n+ b_n)^∞_n=0, a · (a_n)^∞_n=0= (aa_n)^∞_n=0 where a ∈ R. Moreover, we define the product of (aⁿ)^∞_n=0and (bn)^∞_n=0by

(an)^∞_n=0? (bn)^∞_n=0= (cn)^∞_n=0 where c_n=Pn

j=0a_jb_n−j. For any sequence a in V, we define aⁿ inductively by aⁿ= aⁿ⁻¹? a, for n ≥ 1.

Notice that the sequence (a_n)^∞_n=0may be written as (a₀, a₁, a₂, · · · , a_n, · · · ). Hence the prod- uct of (a_n)^∞_n=0 and (b_n)^∞_n=0can be rewritten as:

(a0, a1, a2, · · · ) ? (b0, b1, b2, · · · ) = (a0b0, a0b1+ a1b0, a0b2+ a1b1+ a2b0, · · · ).

(a) Let f = (1, 1, 1, 0 · · · ) and g = (−1, 3, 2, 1, 1, 0 · · · ). Find (−5)f + 2g and f ? g.

(b) Let X = (0, 1, 0 · · · ). Find all Xⁿ for n ≥ 1.

(c) Let X⁰= (1, 0 · · · ). Write f and g in terms of (linear combination) Xⁿ for n ≥ 0.

(d) A polynomial over R is a sequence (aⁿ)^∞_n=0 in V with the property that there exists N ∈ N such that aⁿ= 0 for all n > N.

Let W be the subset of V consisting of polynomials over R. Show that if f, g ∈ W, then f ± g ∈ W and af ∈ W for any a ∈ R.¹Moreover, f ? g ∈ W.²

(e) Let f = (an)^∞_n=0∈ W and assume that an = 0 for all n > N. Here N ∈ N. Prove that f =

N

X

n=0

anXⁿ.

3

(f) Show that if c0, · · · , cN ∈ R such thatPN

n=0cnXⁿ= 0, then c0= c1= · · · = cN = 0.⁴

5

(g) Let f , g ∈ R[[X]]. Prove that if f ? g = 0, either f = 0 or g = 0.

(h) Let f = 1 + X. Find g such that f ? g = 1. (Hint: consider the formal power series 1 + X + X²+ · · · .)

(i) In general, let f = a0+ a1X + a2X²+ · · · be a formal power series over R with a⁰6= 0.

Show that there exists a unique g ∈ R[[X]] such that f ? g = 1. In this case, we write g = f⁻¹ or 1/f .

(j) Let f =

∞

X

n=0

1

n!Xⁿ. Find f^mfor any m ∈ N. Also find f⁻¹ (k) Let f , g ∈ R[[X]]. Prove that for any positive integer n, one has

(f + g)ⁿ =

n

X

k=0

n k



f^k? g^n−k.

(l) Let f =PN

k=0akX^k ∈ R[X]. We say that f has degree N if aN 6= 0. Let g be any other polynomialPM

j=0b_jX^j∈ R[X]. Prove that there exists Q, r ∈ R[X] such that f = g ? Q + r,

where either r = 0 or deg r < deg g.

1This implies that W is a vector subspace of V.

2This implies that the product of polynomials is again a polynomial.

3This is equivalent to say that W is spanned by {Xⁿ: n ≥ 0}.

4This is equivalent to say that {Xⁿ: n ≥ 0} is linearly independent.

5(e) and (f) shows that {Xⁿ: n ≥ 0} forms a basis for W. This is the reason why we usually denote W by R[X]

and V by R[[X]]. Elements of R[[X]] are called formal power series over R. We often denote X⁰ by 1. Elements of R[X] are denoted byP∞

n=0anXⁿ.

1

(m) For each a ∈ R, we denote X − a1 simply by X − a. For any f ∈ R[X] be a polynomial of degree N, show that there exist c1, · · · , cN such that

f =

N

X

i=0

ci(X − a)ⁱ.

6

(n) Let f =PN

i=0c_i(X − a)ⁱ and g =PM

i=0d_i(X − a)ⁱ. Find a formula for f ? g. Is it true that f ? g =PM +N

k=0 e_k(X − a)^k where e_k =Pk

i=0c_id_k−i? (2) Let

f =

∞

X

n=0

(−1)ⁿX²ⁿ (2n)! , g =

∞

X

n=0

(−1)ⁿX²ⁿ⁺¹ (2n + 1)!

be formal power series over R. By (3-g), f⁻¹ exists. We denote f⁻¹? g by g/f . Write g

f =

∞

X

n=0

a_nXⁿ.

Find a₀, a₁, a₂, a₃, a₄, a₅ (The corresponding convergent power series of g/f is in fact the Maclaurin expansion of tan x.)

Definition 2.1. If f is a smooth function in a neighborhood of a, the Taylor polynomial of order n of f (x) at a is given by

T_n(f, a) = f (a) +f⁰(a)

1! (X − a) + · · · +f⁽ⁿ⁾(a)

n! (X − a)ⁿ. The Taylor series generated by f at a is the formal power series:

T_a(f ) =

∞

X

n=0

f⁽ⁿ⁾(a)

n! (X − a)ⁿ.

(3) Suppose f is a differentiable function defined in (−δ, δ) for some δ > 0 such that f⁰(x) = 1 + (f (x))¹⁰, f (0) = 1.

Find T3(f, 0).

(4) Let us define a function f : R → R by f (x) =

(e^−x21 if x 6= 0, 0 if x = 0.

Prove f ∈ C^∞(−∞, ∞) and f⁽ⁿ⁾(0) = 0 for all n ≥ 1 by induction. By definition, the Maclaurin series generated by f is the formal power series T₀(f )(X) = 0. In this example, we see that the Maclaurin series of f does not converge to f (x).

(5) Prove that the following function j(x) =

(

e^{−(1−x2 )21} if x ∈ (−1, 1) 0 if x 6∈ (−1, 1).

is C^∞. Sketch the graph of j(x).

(6) Let A = C^∞(a − d, a + d) be the space of smooth functions on (a − d, a + d). Define Ta : A → R[[X]], f 7→ f⁽ⁿ⁾(a)

n!

^∞

n=0

. Let f, g ∈ A and c ∈ R.

(a) Show that T_a(f + g) = T_a(f ) + T_a(g).

(b) Show that T_a(cf ) = cT_a(f ).

6We can check that {1, (X − a)ⁿ: n ≥ 1} also forms a basis for R[X].

(c) Show that Ta(f g) = Ta(f ) ? Ta(g).

(7) Suppose an ∈ R and

∞

X

n=0

3ⁿan is convergent. Which of the following series is convergent?

(1)

∞

X

n=0

(−2)ⁿa_n, (2)

∞

X

n=0

n(−2)ⁿa_n, (3)

∞

X

n=0

4ⁿa_n.

If it is convergent, prove it. If it is not convergent, give a counterexample.

(8) Find the radius/interval of convergence of the following infinite series.

(a)

∞

X

n=2

(−1)ⁿ xⁿ 4ⁿln n. (b)

∞

X

n=1

n 6ⁿxⁿ, (c)

∞

X

n=1

xⁿ

n^p, where p ∈ R.

(d)

∞

X

n=1

 1 + 1

n

ⁿ² xⁿ.

(e)

∞

X

n=1

(n!)² (2n)!xⁿ. (f)

∞

X

n=1

 1 · 3 · · · (2n − 1) 2 · 4 · · · (2n)

  x − 3 2

n

.

(g)

∞

X

n=0

xⁿ² 2ⁿ . (h)

∞

X

n=0

x^n!.

(i)

∞

X

n=1

 3ⁿ n +5ⁿ

n²

² xⁿ.

(9) Find the Taylor expansion of the following functions at the given point.

(a) sin x, a = π/4.

(b) x sin x, a = π/2.

(c) sinh x, a = 1.

(d) Z x

0

sin t

t dt, a = 0.

(e) Z x

0

tan⁻¹t

t dt a = 0.

(f) Z x

0

sin(t²)dt.

(g) Z x

0

e^−t2dt.

(h) Z x

0

ln(1 + t) t dt.

(10) Find the radius of convergence of the following power series and evaluate their sum.

(a)

∞

X

n=0

x²ⁿ⁺¹ 2n + 1. (b)

∞

X

n=1

xⁿ n(n + 1).

(c)

∞

X

n=1

n(n + 1)xⁿ.

(d)

∞

X

n=1

n²xⁿ.

(e)

∞

X

n=0

x²ⁿ (2n)!. (f)

∞

X

n=1

(−1)ⁿ⁻¹n²xⁿ.

(g)

∞

X

n=1

nx³ⁿ.

(11) Let f (x) = e^xfor x ∈ R. By fundamental theorem of calculus, we know that for x ∈ R e^x= 1 +

Z x 0

e^tdt.

(a) Use induction to prove that for all x ∈ R, e^x= 1 + x

1!+ · · · +xⁿ n! +

Z x 0

(x − t)ⁿ n! e^tdt.

(b) We know that

e^x= 1 + x

1!+ · · · =

∞

X

n=0

xⁿ

n!, for all x ∈ R.

Find a_n such that

e^x− 1 x = 1 +

∞

X

n=1

anxⁿ.

(c) Using an found in (b), we define a formal power series f by f = 1 +

∞

X

n=1

anXⁿ.

Using (3-g), we know that there exists g ∈ R[[X]] such that f ? g = 1. Write g =

∞

X

n=0

B_nXⁿ n! .

Find B₀, B₁, B₂ and all B_2k+1 for k ≥ 1. (The numbers B_n are called Bernoulli num- bers.) The corresponding convergent power series of g is denoted by

x e^x− 1 =

∞

X

n=0

Bnxⁿ n! . Can you find the radius of convergence of g?

(12) Prove that π 4 =

∞

X

n=0

(−1)ⁿ 2n + 1.

(13) Use the binomial series to expand 1/√

1 − x²and find the Maclaurin series for sin⁻¹x.

(14) Evaluate f⁽ⁿ⁾(a) :

(a) Let f (x) = (x − 1)⁷e^x. Find f⁽²⁰⁾(1).

(b) Let f (x) = x sin x for x ∈ R. Find f⁽¹⁰⁰⁾(π/2).

(15) Let F (x) = Z x

0

t

1 + t²dt for x ∈ R.

(a) Find a such that F (a) =

∞

X

k=0

(−1)^k 2k + 2. (b) Evaluate

∞

X

k=0

(−1)^k 2k + 2. (16) (a) Evaluate

Z 1 0

te^tdt.

(b) Evaluate

∞

X

n=1

1 n!(n + 2).

(17) Evaluate the sum of the following series:

(a)

∞

X

n=1

n² 2ⁿ. (b)

∞

X

n=1

n(n + 1) 3ⁿ . (c)

∞

X

n=0

1 (2n + 1)2²ⁿ⁺¹. (d)

∞

X

n=1

n (n + 1)!. (e)

∞

X

n=0

n²+ 3n (n + 2)!.

(18) Use series to evaluate the limit.

(a) lim

x→0

x − ln(1 + x)

x .

(b) lim

x→0

1 − cos x 1 + x − e^x. (c) lim

x→0

sin x − x + ¹₆x³

x⁵ .

(d) lim

x→0

tan x − x x³ .

(19) Use multiplication or division of power series to find the first thre terms in the Maclaurin series for each function.

(a) y = e^−x2cos x (b) y = sec x

(c) y = x sin x (d) y = e^xln(1 + x).

(20) Find the Maclaurin series for the given functions.

(a) f (x) = e^x+ e^2x. (b) f (x) = x²ln(1 + x³).

(c) f (x) = x cos^x₂². (d) f (x) = x

4 + x². (e) f (x) = sin²x.

(21) (a) Show that

Z 1/2 0

dx

x²− x + 1 = π 3√

3.

(b) Using the Maclaurin expansion for 1/(1 + x³) and the above integral to prove that π = 3√

3 4

∞

X

n=0

(−1)ⁿ 8ⁿ

 2

3n + 1+ 1 3n + 2



(22) Let (an) be a sequence of numbers defined by

an+2= 5an+1− 6an, n ≥ 0 with a0= a1= 1. Define a formal power series f =

∞

X

n=0

anXⁿ. (a) Find the radius R of convergence of f .

(b) Show that

f = 1 − 4X 1 − 5X + 6X².

(c) Find the partial fraction expansion of f : i.e. find A, B such that f (X) = A

1 − 2X + B 1 − 3X. (d) Solve for {an}.

(23) Let f =P∞

n=0a_nXⁿ∈ R[[X]] be a formal power series. Recall that the formal derivative of f is defined to be

f⁰=

∞

X

n=1

nanXⁿ⁻¹. Find all f ∈ R[[X]] satisfying the equation

f⁰⁰+ 4f = 0.

Here 0 = (0 · · · ). Find the radius of convergence of f . (24) Prove that

sin⁻¹x

√1 − x² =

∞

X

n=0

2²ⁿ(n!)²

(2n + 1)!x²ⁿ⁺¹, |x| < 1.

(25) In this exercise, we are going to study the following second order linear differential equation:

(2.1) d

dx((1 − x²)y⁰) + n(n + 1)y = 0.

Here n is a natural number. Solutions to this equation are called Legendre function, named after Adrien-Marie Legendre. To find Legendre functions, we use method of power series.

(a) Assume that y =

∞

X

m=0

amX^mis a formal power series satisfying ((1 − X²) ? y⁰)⁰+ n(n + 1)y = 0.

Prove that

a2= −n(n + 1)

2 a0, a3=2 − (n(n + 1))

3! a1

and for m ≥ 2,

am+2= m(m + 1) − n(n + 1) (m + 2)(m + 1) am. (b) Let y1=

∞

X

k=0

a2kX^2k and y2=

∞

X

k=0

a2k+1X^2k+1. Show that

(i) if n is an even natural number, y1 is a polynomial of degree n, and (ii) if n is an odd natural number, y2 is a polynomial of degree n.

(c) Show that the polynomial in (i) and (ii) of (b) is of the form

y = cn [n/2]

X

k=0

(−1)^k 2^kk!

n(n − 1) · · · (n − 2k + 1)

(2n − 1)(2n − 3) · · · (2n − 2k − 1)X^n−2k.

In other words, you need to prove that c_n−2k =(−1)^k

2^kk!

n(n − 1) · · · (n − 2k + 1) (2n − 1)(2n − 3) · · · (2n − 2k + 1)c_n. Here [n/2] is the Gauss-symbol of n/2.

(d) Find c_n such that y(1) = 1 in (c). Denote this polynomial by P_n. Hence Pn= 1

2ⁿ

[n/2]

X

k=0

(−1)^k(2n − 2k)!

k!(n − k)!(n − 2k)!X^n−2k. (e) The Legendre polynomial P_n also has the following expression

P_n= 1 2ⁿn!

dⁿ

dXⁿ(X²− 1)ⁿ. Prove that P⁰_n+1= X ? P⁰_n+ (n + 1)Pn.

(26) Let Pn : [−1, 1] → R be the polynomial function defined by the Legendre polynomial with domain [−1, 1] : (Pn, [−1, 1]).

(a) Show that Z 1

−1

x^mP_n(x)dx = 0 for m = 0, 1, · · · , n − 1.

(b) Evaluate Z 1

−1

Pn(x)²dx.

(c) Evaluate Z 1

−1

x^n+kPn(x)dx for k = 0, 1, 2.