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5*. Consider all sequences (ai,. .., an) such that a, are nonnegative integers and a ai+ 2. Let P...
III. Let k be nonnegative integer. Consider the power series Ord(n + k)! (2) Ja(z) :=...
III. Let k be nonnegative integer. Consider the power series Ord(n + k)! (2) Ja(z) := called the Bessel function of the first kind of order k. Prove that Jk satisfies Bessel's differential equation Hint: You have learned the crucial idea that you can encode interesting recurrence relations via generating functions aka power series. What recurrence relation among the coefficients does this differential equation give?
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
Solve and show work for problem 8 Problem 8. Consider the sequence defined by ao =...
Solve and show work for problem 8
Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for ...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
Let Mi be the set of all sequences {a.);, of real num bers such that Σ converges. More formally, ...
Let Mi be the set of all sequences {a.);, of real num bers such that Σ converges. More formally, we could write this as 1 lal M1a :(W) ai R and i=1 We introduce a function p: Mi x MiR by setting 95 Let (Mi,p) denote the particular metric space we introduced above, and for each X = {xīた1 e M and for each i, we refer to the number xi as the ith coordinate of X. For each N...
Let ao 2 bo > 0, and consider the sequences an and bn defined by an...
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,...,...
real analysis
hint
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,..., defined as follows: X E if and only if II x11 if and only if lxsup lloo. for which ξί (a) If y = {nJ E 11 and Ax = Σ ζίηǐ for every x ε co, then Λ is a bounded linear functional on (More precisely, these two spaces are not equal; the preceding statement exhibits...
Materials: ------------------------------------------------------------------ 9. Let f E (R" where R" is the standard Euclidean space (vector space...
Materials:
------------------------------------------------------------------
9. Let f E (R" where R" is the standard Euclidean space (vector space Rn equipped with the Euclidean scalar product) (i) Explain why there are constants ai,....an R such that 21 ii) Obtain u R" such that f(x)-(1,2), х є R". (ii Explain why the correspondence f u establishedin) is 1-1, onto, and linear so that (R" and R" may be viewed identical. With the usual addition and multiplication, the sets of rational numbers, real numbers, and...
19. Let p be the nth prime number (so pi 2, p2 3, ps 5, and...
19. Let p be the nth prime number (so pi 2, p2 3, ps 5, and so on). (a) Prove that Pr#Q(VP, VP2,.. P-1. [Hint: Use a proof by induction on QVPI VP2, VPn-1) and Fo QPI, VP2VP). (b) Deduce that Q(VPi, VP2 (e) Deduce that Q(PIp 2 2 is a prime) is an Use F-1 , VPn) Q2" for all positive integers . algebraic extension of Q of infinite degree. [This Exercise is motivated by [54].]
III. Let k be nonnegative integer. Consider the power series Ord(n + k)! (2) Ja(z) := called the Bessel function of the first kind of order k. Prove that Jk satisfies Bessel's differential equation Hint: You have learned the crucial idea that you can encode interesting recurrence relations via generating functions aka power series. What recurrence relation among the coefficients does this differential equation give?
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
Solve and show work for problem 8
Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
Let Mi be the set of all sequences {a.);, of real num bers such that Σ converges. More formally, we could write this as 1 lal M1a :(W) ai R and i=1 We introduce a function p: Mi x MiR by setting 95 Let (Mi,p) denote the particular metric space we introduced above, and for each X = {xīた1 e M and for each i, we refer to the number xi as the ith coordinate of X. For each N...
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
real analysis
hint
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,..., defined as follows: X E if and only if II x11 if and only if lxsup lloo. for which ξί (a) If y = {nJ E 11 and Ax = Σ ζίηǐ for every x ε co, then Λ is a bounded linear functional on (More precisely, these two spaces are not equal; the preceding statement exhibits...
Materials:
------------------------------------------------------------------
9. Let f E (R" where R" is the standard Euclidean space (vector space Rn equipped with the Euclidean scalar product) (i) Explain why there are constants ai,....an R such that 21 ii) Obtain u R" such that f(x)-(1,2), х є R". (ii Explain why the correspondence f u establishedin) is 1-1, onto, and linear so that (R" and R" may be viewed identical. With the usual addition and multiplication, the sets of rational numbers, real numbers, and...
19. Let p be the nth prime number (so pi 2, p2 3, ps 5, and so on). (a) Prove that Pr#Q(VP, VP2,.. P-1. [Hint: Use a proof by induction on QVPI VP2, VPn-1) and Fo QPI, VP2VP). (b) Deduce that Q(VPi, VP2 (e) Deduce that Q(PIp 2 2 is a prime) is an Use F-1 , VPn) Q2" for all positive integers . algebraic extension of Q of infinite degree. [This Exercise is motivated by [54].]
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