Sigma notation - mathcentreacuk
We can now see that k-th term is (−1)k 1/k, and that there are 100 terms, so we would write the sum in sigma notation as X100 k=1 (−1)k 1 k Key Point To write a sum in sigma notation, try to find a formula involving a variable k where the first term can be obtained by setting k = 1, the second term by k = 2, and so on Exercises 3
INTRODUCTION TO SIGMA NOTATION
Sigma notation is a way of writing a sum of many terms, in a concise 1 k + 1 Here, the index k takes the values 0, 1, 2, and 3 We’ll plug those each into 1 k+1
Introduction to Section 51: Sigma Notation, Summation
For each natural number k, we de ne p k(n) = Xn j=1 jk; for n = 1;2;::: Then p k(n) is a polynomial in n of degree k + 1, and its leading coe cient is 1 k + 1 In other words: Xn j=1 jk = nk+1 k + 1 + nkc k + n k 1c k 1 + :::nc 1 + c 0; where c 0;c 1;:::c k are some coe cients
Sigma Notation & Sequences - Past Edexcel Exam Questions
Sigma Notation Questions Question 4 - Jan 2011 9 A sequence a 1, a 2, a 3, , is de ned by a 1 = k; a n+1 = 5a n + 3; n 1 where k is a positive integer (a) Write down an expression for a 2 in terms of k [1] (b) Show that a 3 = 25k + 18 [2] (c) i Find P4 r=1 a r in terms of k, in its simplest form ii Show that P4 r=1 a r is divisible by
Sigma Algebras and Borel Sets - George Mason University
Remark 0 1 It follows from the de nition that a countable intersection of sets in Ais also in A De nition 0 2 Let fA ng1 n=1 belong to a sigma algebra A We de ne limsupfA ng= \1 k=1 " [1 n=k A n #; and liminffA ng= [1 k=1 " \1 n=k A n #: Remark 0 2 (1) limsupfA ngis the set of points that are in in nitely many of the A n, and liminffA
USO DEL SIGMA - Recursos Didácticos
Se lee: la sumatoria de los elementos a k desde k = n hasta k = m Nota: 1 Los límites superior e inferior de la sumatoria son constantes respecto del índice de la sumatoria 2 El límite inferior es cualquier número entero menor o igual que el límite superior; es decir, n m Ejemplo: Calcular: 5 k 3 (2k 1) Solución: 1(3)-1 k=3 2(4)-1 k=4
MacroExercicesSupCorrige - maths-francefr
Title: MacroExercicesSupCorrige dvi Created Date: 1/24/2016 4:54:59 PM
Chapter 1
• Sigma algebras can be generated from arbitrary sets This will be useful in developing the probability space Theorem: For some set X, the intersection of all σ-algebras, Ai, containing X −that is, x ∈X x ∈Ai for all i− is itself a σ-algebra, denoted σ(X) This is called the σ-algebra generatedby X Sigma-algebra Sample Space, Ω
Continuous-Time Delta-Sigma Modulators: Tutorial Overview
time delta sigma modulators (DTDSMs) are recommended to see another tutorial paper [8] 2 Fundamentals of Delta-Sigma Modulators 2 1 Basic Characteristics Figure 1 shows the basic concept of the delta-sigma modula-tor The modulator is a kind of filter system for quantization noise by using a feedback loop
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INTRODUCTION TO SIGMA NOTATION
1.The notation itselfSigmanotation is a way of writing a sum of many terms, in a concise
form. A sum in sigma notation looks something like this: 5 X k=13k The (sigma) indicates that a sum is being taken. The variablek is called theindex of the sum. The numbers at the top and bottom of the are called theupperandlowerlimits of the summation. In this case, the upper limit is 5, and the lower limit is 1. The notation means that we will take every integer value ofkbetween 1 and 5 (so1, 2, 3, 4, and 5) and plug them each into the summand formula (here
that formula is 3k). Then those are all added together. 5 X k=13k= 31 + 32 + 33 + 34 + 35 = 45 Example 1.Write out what is meant by the following: 3 X k=01k+ 1 Here, the indexktakes the values 0, 1, 2, and 3. We'll plug those each into1k+1and add them together.
3 X k=01k+ 1=10 + 1 +11 + 1 +12 + 1 +13 + 1 Example 2.Write out what is meant by the following: 8 X i=1(1)iDate: Friday March 31, 2016. 12 INTRODUCTION TO SIGMA NOTATION
The index variable here is written asiinstead ofk. That's ok. The most common variables to use for indexes includei,j,k,m, andn. 8X i=1(1)i= (1)1+ (1)2+ (1)3+ (1)4+ (1)5+ (1)6+ (1)7+ (1)8 =1 + 1 +1 + 1 +1 + 1 +1 + 1 = 0Try one on your own.
Example 3.Write out what is meant by the following (no need to simplify): 4X n=1pn+ 1Let's try going the other way around.
Example 4.Write the following sum in sigma notation.2 + 4 + 6 + 8 +:::+ 22 + 24
Notice that we can factor a 2 out of each term to rewrite this sum as21 + 22 + 23 + 24 +:::+ 211 + 212
That means that we are adding together 2 times every number between1 and 12. The sigma notation could be
12X k=12k There is no need to usekas our index variable. We could have just as easily usedmorjinstead. 12X k=12k=12X m=12m=12X j=12jNotice, that these are NOT the same as
12X k=12m Example 5.Write the following sum in sigma notation. 112+14 18 +:::164 +1128
INTRODUCTION TO SIGMA NOTATION 3
This one is a little more complicated. We'll worry about the signs later, rst we'll deal with the numbers themselves. Do you notice a pattern in the terms? Sure, we get to the next term by dividing by 2.That is:
1 = 12 0=12 0 12 =12 1=12 1 14 =12 2=12 2 1128=12 7=12 7 If we call our index variablek, thenkshould go from 0 to 7, and the numbers themselves are just12 k. Now we need to deal with the signs. We say above that (1)kwill alternate between +1 and1. That is, if we multiply our terms from above by (1)k, they will alternate between + and. We are starting withk= 0, so (1)0= +1 will give us the alternation starting at the sign we want. 7X k=0(1)k12 k =7X k=0 12 k