1 {a,}={ 4. Let | 2n² Let ɛ =.001. Find a natural number N such that |a, - L<ɛ when n 2 N.

Q: 13.Show that the third-order Adams-Moulton formula is yn+1=yn+h12(5fn+1+8fn−fn−1).

A:

Q: A sequence sn is said to be bounded above if there is some number N such that %3D Sn SN.cO S,2N.dO

A: Bounded Above Sequence: A sequence an is said to be bounded above if there exists a real number M…

Q: a=t). CTaylon Serie) 2-X Ln +x) (MacLurinSeries

A: We need to find the Taylor series of f(x)=12-x at a=1. Taylor series of function fx at a is defined…

Q: Assume a; > 0 (i = 1,..., n). Show that 1 +. a1 < (a1a2 ... an)/n. n an

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Q: The results of a mathematics exam at JUST university for two cla Class 1: (n=32, mean=15.7, SD=8)…

A: Given,n1=32 , μ1=15.7 , σ1=8n2=36 , μ2=4.5 , σ2=12α=1-0.95=0.05α2=0.025Z0.025=1.96

Q: Let xn ≥ 0 for all n ∈ N. (a) If (xn) → 0, show that (√xn) → 0. (b) If (xn) → x, show that (√xn)…

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Q: - - 9. - [1.0001 x* H. Verify Let A 6 = the error %3D 1.0001 formula |r* – ô|| ||r|| < K(A)-

A: Let A=2112, b=33, x^=1.00011.0001, x*=11. Verify the error formula x*-x^x*≤KArb.

Q: Let U1, U2, ... be i.i.d. Uni(0, 1) random variables. - log U; is distributed as exp(1). (2) Find…

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Q: 2. Use the Root Test to determine absolutely or diverge. 4n+3\ n (a)E-1( Zn=1 Зп-5) (b) E (-In (e²…

A: As we have to use the root test , So first I have written the Root test then using them for both the…

Q: Assume that after w months,butterfiles populace in 1000 is calculated as F(w)=12w+cos(16w)+n×100…

A: Minimum and maximum of a function depends upon all the variables present in the function.

Q: 3 E in (1 -) -in2 , . =-In2 =1 k2 –1 k=2

A: We will  use the limit of partial sum

Q: a) Show that E-o(-1)*(")=0, and then verify that () + C) + C') + -- - C) + (") + (") + -- .. b)…

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Q: If (n + 1)(n + 2)an+1x" - nan-1x" = 0, n=1 show that for k = 1, 2, 3, ..., we have 1.3.5... (2k - 1)…

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Q: Given: In(1+x) = >) (-1)". -xn+1 if |x| < 1. n+1 n=0 Compute the exact value of -1)"(n+2) 2n+1(n +…

A: Given: ln(1+x)=∑n=0∞(-1)nn+1xn+1 where x&lt;1To Compute: Exact value of ∑n=0∞(-1)n(n+2)2n+1(n+1)

Q: * :The nth term formula for { 1,0,1,0,... } is 1-(-1)" 2. an = A₂ = An خيار 2 1 + (−1)" 2.

A:

Q: 3 In Oregon, of the soft drink bottles are recycled each year. A bottling company there produces 1.2…

A: Given, 35 of the soft drink bottle are recycles each year and x year later the number of bottles…

Q: I    P    R    T ----------    2000    2%    ½ yrs. 500    ----------    10%    7 mos. 175    1030…

A: According to guidelines I solve first 3 subparts so kindly request you to repost the remaining…

Q: Let a, b eR with a < b. Assume that a + <b - for all n e N. Show that n n=1

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Q: Suppose that there are currently unemployed workers în some state. Each mðnth 8% of all those…

A: Given data: Number of unemployed workers=25,000 Let x be the number of months. Given that each month…

Q: Show that 100 1 arctan(100) – arctan(2) + 1 1 < arctan(100) – arctan(2) + 1+ 1002 1+ k2 k=2 1+ 22

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Q: 1. Find two positive numbers such that their product is 192 and the sum of the first and thrice the…

A: We have to use the application of derivatives to solve this given problem

Q: P-value = 0.072, a = 0.05 O reject Ho O do not reject Ho P-value = 0.005, a = 0.001 O reject Ho O do…

A: We need to determine the decision based on the given p-values and significance level.

Q: If In x + In(x – 2) = ln(7x), then x -

A: From the given problem:   lnx+lnx-2=ln7x   Now using the logarithmic property:   ln(a·b)=ln(a)+ln(b)

Q: The interval [0,5] is partitioned into nn equal subintervals, and a number xi is arbitrarily chosen…

A: The given interval is [0,5]. Find the xi.

Q: 3 If Xn 4 and ly,l0. Select one: True O False

A: Given, xn → 4  and  yn≤31n2n for all values of n To determine whether ∑ xnynp  is absolutely…

Q: Find the limit of the sequence bn = (cos n)/(n2 + 1).

A: Given sequence is bn=cosnn2+1.

Q: Show that 100 1 arctan(100) – arctan(2) + E s arctan(100) – arctan(2) + 1+ 1002 1+ k2 k=2 1+ 22

A: To show that: arctan100−arctan2+11002≤∑k=210011+k2≤arctan100−arctan2+122

Q: 16. There is a very quick way to compute £-, 2'(-1)ª-' (); do use it. l=D0

A: Introduction: The binomial theorem is very important in mathematics. For any real numbers a,b with…

Q: 2. (a) For n ≥ 0 and x E R, prove that (b) Prove n n Σ ^ (^.) 2^* (1 - 2)"-* k k=0 = = In. n x)n-k…

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Q: Calculate dIn |2 – 1|/d In a as a function of 2 and w; by differentiating 2(a) – 1 = H²a? to show d…

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Q: 4. Compute the exact value of (-1)"(n+2) xn+1 Ln=0 2n+1 (nt1) 100

A: The geometric series formula is ∑n=0∞rn=11-r, where r&lt;1. Break the numerator of the given…

Q: () 2k = 3", Vn E N k k=0 k n+1 Σ 2 Vn E N, n > 1 3 k=2 į * () 主) n 3k = 4", Vn E N %3D k k=0 = 2"-1,…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2 For n=1 a) Verify that that lin b₁=0. 7380 b) Compute Sy. <) Find an d) How малу Must be added Sum…

A: This is a multipart question having three or more interrelated subparts. As per the company policy,…

Q: 1. For an = 5n2 + 9n3 + 1, an = n2, prove or disprove: rn = O(an) (n → o)?

A: We will use the standard definition.

Q: 8. If l is the output for Carmichael's function machine shown at right, how can you determine what…

A: Since the output for Carmichael's function machine is 1. Equate x2+2x+1 to the output value of 1 and…

Q: {Xn} 3. Prove that the sequence 1+2n² 1-3n² Xn where is bounded.

A: As per the company's guidelines, we are supposed to solve first question for you, kindly repost…

Q: For xn = 5n² + 9n³ + 1, an = en = n², prove or disprove: an = O(an) (n → 0)?

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Q: Find the limit of the following sequence: 00 cos(2n? + 10) (-1)"-ln3 n=1

A: The objective is to find the limit of the following sequence.

Q: 5. Prove that . Hint: Compute e-' from the continued frac e - 1 1 + 2 + 3 + ·…. tion for et given in…

A: When a series is in the form ∑k=1∞-1k-1α1α2α3...αk, we can express it as a continued fraction as…

Q: Let b₁ 2 = sin−¹ (²) and c, = sin-¹ („²7) for all n € N such that n ≥ 2 +00 Σ(1+bn) n=2 Σ(bn-Cn) n=2…

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Q: 5. (8x*y® + 3x³) dx + (12x°y³ + 3xy²) dy = 0

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Q: 3. Let, a1 = 3,az = 4 and for n > 3, an = 2an-1+ an-2 – 5, express a, in terms of n. %3D %3D

A: An arithmetic progression is a sequence of numbers such that the difference d between each…

Q: Find the nth term of (a, b, a, b ... ), (a) and .(b) are constants 1-(-1)" 1- (−1)n+1 -[-D" + 2 2 a…

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Q: 1) Prove that 2" 4 2) Prove that 4 divides 7" -3" for eachn E N 3) Prove that | sin na| 3 f(n) =…

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Q: 5. Show that cos .3 G) do not converge.

A: We need to prove that: cosnπ3 does not converge This can be proved as follows:

Q: 6. Given that -뿔 (금) (규 R(x) 4! 1+. for x E (-, 3), where & is between x and 0, find an upper bound…

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Q: (-1) is where n is an integer. equal to the above e(1+2n)i where n is an integer. equal to the abov…

A: We know from eular formula e^iθ = cos θ + isin θ

Q: Two real numbers X and Y are chosen at random in the interval (0, 1). Compute the probability that…

A: Selecting two points x and y from the interval (0,1) is identical to selecting a single point from…

Q: Suppose the interval [1,9] is divided into n subintervals of equal width, and Ck represents the left…

A:

Ratios

A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.

Trigonometric Ratios

Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.