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Real Analysis

Sequences and Series (20 Problems): Partial Sums, Convergence and Divergence

Please do all problems below step by step showing me everything. Do simply as possible so I can clearly understand without rework. Adult here relearning so show all work, etc. OK, some said cannot read problems, but do not have a scanner with me know, so typed them in below. Sorry for any problems, but this shopuld clear up

Finding Limits and Applying the L'Hopital's Rule

Find the indicated limit make sure you have a indeterminate form before you apply L'Hopital rule (1) lim xgo to0 arctan3x/arcsinx (2) lim x go to pi/2 3secx+5/tanx (3)lim x go to 0 2csc^2x/cot^2x evaluate dx/squrtpix a=0 andb=inifinity.

Convergence Tests for Series : Comparison test

Use only the comparison test, limit comparison test, integral test, and nth term test to test for convergence or divergence infinite series(n=2 to infinity) sqrt(25n+4/n^3-n) infinite series (n=2, infinity) ln(n)/n^3 infinite series(n=1, infinity) (n*arctan(2n))/(2n-1) *=multiply

Series: Absolute Convergence

Suppose the summation from k=1 to n of a_k is absolutely convergent and {b_n} is bounded. Prove that this implies the summation from k=1 to n of a_k*b_k is absolutely convergent.

Cauchy Sequence and Limit Supremum

Suppose that {a_n}_n is a real Cauchy sequence. Prove that lim superior n--> infinity (a_n) = lim inferior n--infinity (a_n) so as to conclude that lim n--infinity (a_n) exists.

Limit Proofs

Prove the following a) If lim n-->infinity (a_n*b_n) exists and lim n--> infinity (a_n) exists, then lim n -->infinity (b_n) exists. b) If lim n--> infinity (a_n) = 0 and {b_n} is bounded, then lim n-->infinity (a_n*b_n) exists and equals 0. c) If lim superior (a_n) exists, then {a_n}_n is bounded above.

Sequences and Limits

Consider the real sequence {x_n}_n generated by the iteration scheme x_n+1 = x_n(2-ax_n), for n = 0, 1, 2, ...... where a>0 and x_0 satisfying 0 < x_0 </= 1/a. a. Prove 1/a>/=x_n>0 for all n. b. Prove x_n>/=x_n-1. c. Conclude that lim n-->infinity

Limit Inferior Functions

Suppose a_n >0 for each n in N and lim inf (a_n) > 0. Prove there is a number a>0 st a_n >/= a for all n in N. (limit n--> infinity)

Series : Domain of Convergence

Find the domain of convergence of each of the following: a) Summation from n = 1 to infinity of [(z-1)^2n]/((2n)!) b) Summation from n = 1 to infinity of [(n^2 + 1)/(2n + 1)]z^n c) Summation from n = 1 to infinity of [(z + 1)^n]/n d) Summation from n = 0 to infinity of [(z - 1)/(z + 1)]^n

Arithmetic Series : Finding d, the difference between any 2 terms.

Use the arithmetic series of numbers 1, 3, 5, 7, 9,...to find the following: What is d, the difference between any 2 terms? Please show me how you get the answers you got and any helpful resources. Using the formula for the Nth term of an arithmetic series, what is 101st term? Using the formula for the sum of an arithme

Dense Subset, Continuity and Uniform Convergence

Let E C R1 and let D be a dense subset of E. If are continuous real-valued functions on E for n=1,2,..., and fn converges uniformly on D, prove that fn converges uniformly on E. (See attached file for full problem description) I am using the book Methods of Real analysis by Richard Goldberg.

Prove that the Series of Functions Converges Uniformly

(See attached file for full problem description with equations) --- 9.3-5 Let {f_n} (from n - 1 to infinity) be a sequence of functions on [a,b] such that (f_n)'(x) exists for every x is an element of {a,b](n is an element of I) and (1) {(f_n)(x_0)} (from n=1 to infinity) converges for some x_0 is an element of [a,b]. (2

Methods of Real Analysis by Richard Goldberg

(See attached file for full problem description with equations) --- 9.5-3 Without finding the sum of the series Show that --- We use the book Methods of Real Analysis by Richard Goldberg.

Prove that converges uniformly on E

(See attached file for full problem description with equations) --- 94.8 Let be a sequence of functions on E such that where . Let be a nonincreasing sequence of nonnegative numbers that converges to 0. Prove that converges uniformly on E (Hint: See 3.8C) Theorem 3,8C Let be a sequence of real numbers whos

Methods of Real Analysis by Richard Goldberg.

(See attached file for full problem description with equations) --- 9.4-5 Show that the series is uniformly convergent on [0,A] for any A>0. Prove that --- We are using the book of Methods of Real Analysis by Richard Goldberg.

Limit Sequence Functions

(See attached file for full problem description with equation and proper symbols) --- 9.2-10 If be a sequence of functions that converges uniformly to the continuous function , prove that ---

Sequences and Uniform Convergence

Let {fn} infinity-->n-1 be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each n&#1028; I let Fn(x) = &#8747; x--> a fn(t)dt a<x<b Show that {fn} infinity-->n-1 converges uniformly on [a,b]. (Hint: Use 9.2F) Theorem 9.2F;

Totally Bounded : Let M1 be a totally bounded metric space

6. Let M1 be a totally bounded metric space, and f: M1 --> M2 is uniformly continuous and onto. Show M2 is totally bounded. Note: we are using the "Methods of Real Analysis by Richard R Goldberg" Please see the attached file for the fully formatted problems.

Mapping, Contraction and Fixed-Point Theorem

(See attached file for full problem description with proper equations) --- 3. Let T(x) = x^2 Show that T is a contraction on (0, 1/3] , but that T has no fixed point on this interval. Does this conflict Theorem 6.4? Explain. Note: We are using the book Methods of Real Analysis by Richard R. Goldberg. This

Metric Space and Countable Dense Subset

2. Prove that if a metric space M is totally bounded, then there is a countable dense subset of M. Note: we are using the "Methods of Real Analysis by Richard R Goldberg

Radius and disk of convergence

Find the radius of convergence of the series sum from n = 1 to infinity of n^3(z/3)^n. Does this series converge at any point on the boundary of the disk of convergence?

Limits of a Square Root

As x approaches -2 from the left, what is the limit of (square root of x^2 +5) / (x+2) Please show work if you can. Choices are A. 3/2, B. 0, C. -infinity D. -1, E. + infinity