2. Let {sn}=1 be an unbounded sequence of negative numbers. Show {sn}n=1 has a 00 00 subsequence {Sn, such that {Sng } tends to minus infinity k=1 k=1

Q: Find the sum of the following convergent series. 1 b. 2n=122n-1 1 e 2n+1 (

A: The solution is :

Q: 11. If {an} and {b„} are sequences of real numbers, lim,0 an = 5, lim, bn = 8, and fn(x) = a, + bnx,…

A:

Q: Show that the series Ln=l 12+2z10n-5z"ª –3z+2;10n converges absolutely for |리 < 1.

A: The series ∑n=1∞an is absolutely convergent if the series ∑n=1∞an  Ratio test:  The series ∑n=1∞an…

Q: The series Σ 6( − 1)*0.2-1.9/k○ diverges converges. k=0

A: We use ratio test to find the desired result.

Q: Find the sum of the convergent series.

A: The given question is taken from the sequence and series in which we have to find the value of the…

Q: Let (s) be a sequence of real number, we say s, is Decreasing if for each nEN Sn >Sn+1

A: Given a statement

Q: 21. Let (un) be a decreasing sequence tends to zero. For all n E N, Sn E(-1)*ux- k=0 Show that two…

A: Definition: Two sequences are said to be adjacent sequences, if one of the sequence converge and one…

Q: 5. Find all values of x >0 so that ) converges. 3k k=2

A:

Q: a Let (n) be a constant sequence in R (i.e. there is some constant c such that In = c for all n).…

A:

Q: 4.) Find the sum of the convergent series 4 n(n + 2)

A:

Q: Suppose that (an)n20 is a bounded sequence of real numbers. For each natural number n, define fn(x)…

A: We are given a sequence of functions, where {an} is a bounded sequence. We need to establish the…

Q: Which of the following is true about the sequence In (3n² − n² + 1) − ln(n³ + 2n¹ + 5n)? = an The…

A:

Q: 7. Suppose A and B are compact subsets of R and that fn is a sequence of functions converging…

A:

Q: 21. Show that E(-1)"+1; 2n converges n! absolutely. n=1

A:

Q: Find the sum of the convergent series.

A:

Q: The following series Σ 3n-2" 5n+1 n=0 is convergent to

A: this is convergent series and

Q: The series 7 2n E,(-1)" 4 n= is absolutely convergent. Select one: True O False

A: Root Test : Suppose we have series ∑ an .Define,                          L = limn→∞ an1n .Then ,(…

Q: Show that E-2 ((In n)ª/n²) converges for -o 1. n=2

A:

Q: The sequence {Pn =}1 converges n-1 to p = 0 of order a = %3D O a. 3 O b. 1 О с. 2

A: For some number p, if the condition limn→∞an+1anp=M&gt;0 is satisfied, then the order of convergence…

Q: Let (xn) be a bounded sequence and let s = sup{xn : n e N}. Show that if s ¢ {xn : n E N}, then…

A:

Q: 42n The given power series :)n (x +7)" is convergent if n=0 |(x +7)| < (Write your answer up to 5…

A: Simplify the given series than take the help of sum geometric series concept.

Q: 5. (a) Suppose 1 and Xn+1 = 2 Show that {n} converges and find the limit. X1 for all nEN.

A:

Q: The series k=1 5( − 1)k+1 - kl.1 O converges diverges.

A:

Q: 1.2 Determine whether or not the series of functions {S„(x)} *-1} converges uniformly k=1 on [-a, a]…

A:

Q: Consider the series > (2k + 1) and the sequence {Sn} of partial si k-1 S101 – S98

A:

Q: 8. 3. Use two different methods to show that the series 1 converges. 1+ n2 n=1

A:

Q: The sequence {(1 + 1)^² +1} 2n O O converges to e² is divergent ez converges to e 2 O converges to e

A: Sequence

Q: .8.8 Suppose that Cn >0 for each n E N. Prove that n either converges = diverges to infinity.

A:

Q: 9. The series 3k+ (k+1)* is Convergent Divergent

A: We will solve the following.

Q: ind the sum of the convergent series. +00 3n+1 Z 6n-1 n=0

A:

Q: Q2/ Let x(n) be a 4-point sequence. Compute the DFT of x(n). 1 x (n) =- 0<n<3 0otherwise ans. X(k)…

A: …

Q: If > an converges, then a converges. n=1 n=1

A:

Q: Find the sum of the convergent series. 3 8 n = 0

A:

Q: Consider the sequence S n²r(1 – nx) x € [0, 1/n] - fn(x) := 0. elsewhere . Show it converges…

A: Let gx=n2x1-nx Differentiating wrt x, we get  g'x=n2-2n3x Putting g'x=0, we get  n2-2n3x=0⇒x=12n

Q: 00 n³n² COS 5n³ +3, 1

A: See the answer below

Q: IV. Determine if the given is convergent or divergent. s In"(n*) 23n2 #4. n=1

A: Given series is ∑n=1∞lnn(n4)23n2

Q: Let {Sn} n=1 to infinity, be an unbounded sequence of negative numbers. Show {Sn} n=1 to infinity,…

A:

Q: Find the sum of the convergent series. 1 n vn + 1. n=16

A:

Q: Find the smallest value of x for which the series (エ+48)” 2!n(n) n=2 is conditionally convergent.

A: Given series is ∑n=2∞(x+48)n2ln(n)

Q: Find the values of the convergent series [from k=4 to infinity] E (2/3)3n-1 (-4/7)2n+3 *that E just…

A: Given information:The given convergent series is

Q: If |Xn+2 − Xn+1| < |¤n+1 − xn| for all n E N then (xn) converges. -

A:

Q: Suppose {fn}n=1 and {gn}n=1 converges uniformly on E where {fn}n=1 and {gn}n=1 are sequences of…

A:

Q: 5) Find all values of x for which En=1 nxn (n+1)(2x+1)n converges absolutely.

A:

Q: 21. Find the sum of the convergent series. n =0

A: ∑n=0∞434n We can simplify, = ∑n=0∞22.2-2n3n = ∑n=0∞22-2n3n =4∑n=0∞2-2n3n  = 4∑n=0∞34n

Q: an converges for some large k, then so will n=1 an- If E n=k

A: We have to check

Q: 42n The given power series (x + 8)" is 9n n=0 convergent if |(x + 8)||

A:

Q: 7. Let x₁ = a > 0 and Xn+1 = x + 1/x,, for n E N. Determine whether (x,) converges or diverges.

A: Q7 asked and answered.

Q: 1 Show that the series 2 converges. k=2 k(log k)?

A: Given: ∑k=2∞1klogk2

Q: Find the sum of the convergent series. 6 Σ 9n2 + 3n – 2 n = 1

A:

Q: Determine the one dimensional DFT for the following sequence x(n)= { 0,\ / 7 / l }

A: Sol:- Given Sequence :- xn=0,1,7,1One Dimension DFT  is given by Xw=11111-j-1j1-11-11j-1-j0171

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.