Recent questions tagged sequence

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Define a Cauchy sequence and state its importance in real analysis.
103
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Is there a sequence of polynomials $P_n$ such that:$$\lim_{n\to\infty} P_n(z) = \begin{cases} 1 & \text{ if } \Im(z) 0 \\ 0 & \text{ if } \Im(z) = 0 \\ -1 & \text{ ...
149
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Is there a sequence of polynomials $P_n$ such that $P_n(0) = 1$ for all $n$, but $P_n(z) \to 0$ as $n \to \infty$?
101
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Let $L_1, L_2$ be lines in the plane. For which pairs of $L_1, L_2$ do there exists real functions, harmonic on the entire plane, 0 on $L_1 \cup L_2$, but not vanishing i...
75
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Let $\Omega$ be a region, and $f_n \in \mathcal{H}(\Omega)$ for all $n$. Set $u_n = \Re(f_n)$, and suppose $u_n$ converges uniformly on compact subsets of $\Omega$ and th...
87
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Let $u,v$ be real harmonic functions on a plane region $\Omega$. Under what conditions is $uv$ harmonic?Further, show that $u^2$ may not be harmonic on $\Omega$, unless $...
67
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Compute$$\int_0^\infty \frac{dx}{1 + x^n}$$for $n \geq 2$.
119
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(1) The solution of the inequality \( 3(2-x) \geq 2(1-x) \) for real \( x \) is
97
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Solution \( e^{i \pi}+1=0 \)
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Explain \( e^{i \pi}+1=0 \)
68
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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...
123
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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 \( a=1 \).
83
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Show that a sequence that converges cannot have two limits.
102
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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 \( e^{i \pi}+1=0 \)
92
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What is the arithmetic mean?
75
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What is the palindrome of 66?
164
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What is arithmetic geometry?
131
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What role do incentive schemes play in OOM?
155
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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).
156
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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?
237
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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...
401
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What is the impact of creatine on cognition?
203
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Evaluate \(9^{\frac{ 150}{300}}\)
152
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In what sequence do cognition, affect, and behavior play a role in advertising?
594
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Prove that \(n^2=(n+1)(n-1)+1\)
322
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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, ...
251
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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...
412
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Who discovered pi?
295
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Who discovered this equation \( e^{i \pi}+1=0 \)?
520
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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 ?
317
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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...
210
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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?
282
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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
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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...
441
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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 \(4+\frac{4}{3}+\frac{4}{9}+\ldots\)
440
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The 3rd and 12th terms of an arithmetic progression are 12 and \(-24\) respectively. Determine the 49th term.
524
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The 11th term of an arithmetic sequence is 32 and the sum of the first 11 terms is 187 . Calculate the 21st term.
491
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\(2 ; x ; y\) 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...
319
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Calculate \(n\) if \(\quad \sum_{k=1}^n(3 k+1)=650\)
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