# Recent questions tagged sequence

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Define a Cauchy sequence and state its importance in real analysis.
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Is there a sequence of polynomials $P_n$ such that:$$\lim_{n\to\infty} P_n(z) = \begin{cases} 1 & \text{ if } \Im(z) 0 \\ 0 & \text{ if } \Im(z) = 0 \\ -1 & \text{ ... 149 views 1 answers Is there a sequence of polynomials P_n such that P_n(0) = 1 for all n, but P_n(z) \to 0 as n \to \infty? 101 views 1 answers Let L_1, L_2 be lines in the plane. For which pairs of L_1, L_2 do there exists real functions, harmonic on the entire plane, 0 on L_1 \cup L_2, but not vanishing i... 75 views 1 answers Let \Omega be a region, and f_n \in \mathcal{H}(\Omega) for all n. Set u_n = \Re(f_n), and suppose u_n converges uniformly on compact subsets of \Omega and th... 87 views 1 answers Let u,v be real harmonic functions on a plane region \Omega. Under what conditions is uv harmonic?Further, show that u^2 may not be harmonic on \Omega, unless ... 67 views 1 answers Compute$$\int_0^\infty \frac{dx}{1 + x^n}for $n \geq 2$.
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(1) The solution of the inequality $$3(2-x) \geq 2(1-x)$$ for real $$x$$ is
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Solution $$e^{i \pi}+1=0$$
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Explain $$e^{i \pi}+1=0$$
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Let $$\left\{f_{i}\right\}$$ be a uniform Cauchy sequence consisting of uniformly continuous functions on a closed, bounded interval $$I$$. Show that the limit is uni...
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Show that for any $$0<a<1$$ the sequence $$\left\{x^{i}\right\}$$ converges uniformly to 0 on $$[0, a]$$, but not for $$a=1$$.
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Show that a sequence that converges cannot have two limits.
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Show that a sequence that converges is a Cauchy sequence. Hint: write $$\xi_{i}-\xi_{j}=\left(\xi_{i}-\xi\right)+\left(\xi-\xi_{j}\right)$$ and use the triangle inequal...
104
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The HCF of the two numbers is 29 and their sum is 174. What are the numbers?
122
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Why is graph of sequence discrete?
76
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Evaluate $$e^{i \pi}+1=0$$
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What is the arithmetic mean?
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What is the palindrome of 66?
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What is arithmetic geometry?
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What role do incentive schemes play in OOM?
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Given a sequence of numbers:$2 ;-7 ;-16 ;-25 ;-34 ; \ldots$Determine the common difference for the sequence (if there is one).
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Given a sequence of numbers:$-2 ; 2 ;-11 ;-8 ;-13 ; \ldots$Determine the common difference for the sequence (if there is one).
276
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How do I show that all terms in a pattern are even?
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Consider the following quadratic number pattern: $$6 ; 10 ; 18 ; \ldots \ldots$$1. Write down the following two terms of the pattern.2. Determine the equation of the ge...
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What is the impact of creatine on cognition?
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Evaluate $$9^{\frac{ 150}{300}}$$
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In what sequence do cognition, affect, and behavior play a role in advertising?
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Prove that $$n^2=(n+1)(n-1)+1$$
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Explain $$e^{i \pi}+1=0$$
172
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Find the nth term of the quadratic sequence 2, 6, 12, 20, 30, ...
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Here are the first 5 terms of a sequence. $\begin{array}{lllll}29 & 25 & 21 & 17 & 13\end{array}$Find an expression, in terms of $$n$$, for the $$n$$th term of this seq...
273
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412
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Who discovered pi?
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Who discovered this equation $$e^{i \pi}+1=0$$?
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What is a geometric sequence?
533
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What are axioms of arithmetic?
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What is geometric thinking ?
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$$8 ; 24 ; 72 ; \ldots \ldots$$a) Write down the next 3 terms of the sequence.b) Find the general term.c) Show by calculation whether 5832 is a term of this sequence or n...
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Given the number pattern $$50 ; 47 ; 44$$;a) Determine the nth term of this pattern.b) Determine which term of the pattern is equal to $$-67$$.c) Calculate the number of ...
611
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What is a Cauchy-Riemann equation?
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Evaluate:(1) $$\dfrac{\sin 35^{\circ} \cdot \cos 35^{\circ}}{\tan 225^{\circ} \cdot \cos 200^{\circ}}$$ (2) $$\dfrac{\sin 36^{\circ}}{\sin 12^{\circ}}-\dfrac{\cos 36^{\ci... 410 views 0 answers In an arithmetic sequence \(S_n=n^2-2 n$$. Use your formula in 2 to determine the value of:(a) $$\mathrm{T}_7$$(b) $$\mathrm{T}_{50}$$(c) $$\mathrm{T}_n$$
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Consider the geometric sequence: $$\quad 8 x^2 ; 4 x^3 ; 2 x^4 ; \ldots \ldots \ldots$$(a) Show that the general term is given by $$\mathrm{T}_n=2^{4-n} \cdot x^{1+n}$$(b...
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What is the greatest value of $$p$$ for which the series $$\sum_{k=1}^p 10(3)^{k-1}<2300$$ ?
358
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Calculate the difference between $$S_{\infty}$$ and $$S_7$$ of $$4+\frac{4}{3}+\frac{4}{9}+\ldots$$
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The 3rd and 12th terms of an arithmetic progression are 12 and $$-24$$ respectively. Determine the 49th term.
$$2 ; x ; y$$ are the first three terms of an arithmetic sequence. If the 2nd term is decreased by 1 , the three terms will form a geometric sequence. Calculate $$x$$ and...
Calculate $$n$$ if $$\quad \sum_{k=1}^n(3 k+1)=650$$