MATEMATIKA REKAYASA IMO 141202

Kuliah 4: Deret

mahmud mustain

Materi Kuliah 4Barisan dan Deret Tak Hingga:barisan tak hingga,deret tak hingga,test konvergensi,deret Taylor,deret MacLaurin

Arithmetic Sequences & Serieswebtech.cherokee.k12.ga.us/sequoyah-hs/math/11.2%20Arithmetic%20Sequences%20&%20Series.ppt

Arithmetic Sequence:The difference between consecutive terms is constant (or the same).The constant difference is also known as the common difference (d).(Its also that number that you are adding every time!)

How do I know if it is an arithmetic series?A series is the expression for the sum of the terms of a sequence, not just what is the next termsEx: 6, 9, 12, 15, 18 . . . This is a list of the numbers in the pattern an not a sum. It is a sequence. Note it goes on forever, so we say it is an infinite sequence. Ex: 6 + 9 + 12 + 15 + 18Note: if the numbers go on forever, it is infinite; if it has a definitive ending it is finite. Here we are adding the values. We call this a series. Because it does not go on forever, we say it is a finite series.

Example: Decide whether each sequence is arithmetic.-10,-6,-2,0,2,6,10,-6--10=4-2--6=40--2=22-0=26-2=410-6=4Not arithmetic (because the differences are not the same)5,11,17,23,29,11-5=617-11=623-17=629-23=6

Arithmetic (common difference is 6)

Rule for an Arithmetic Sequencean=a1+(n-1)d

Example: Write a rule for the nth term of the sequence 32,47,62,77, . Then, find a12.The is a common difference where d=15, therefore the sequence is arithmetic.Use an=a1+(n-1)d an=32+(n-1)(15) an=32+15n-15 an=17+15n

a12=17+15(12)=197

Example: One term of an arithmetic sequence is a8=50. The common difference is 0.25. Write a rule for the nth term.Use an=a1+(n-1)d to find the 1st term!a8=a1+(8-1)(.25)50=a1+(7)(.25)50=a1+1.7548.25=a1* Now, use an=a1+(n-1)d to find the rule.an=48.25+(n-1)(.25)an=48.25+.25n-.25an=48+.25n

Example: Two terms of an arithmetic sequence are a5=10 and a30=110. Write a rule for the nth term.Begin by writing 2 equations; one for each term given.a5=a1+(5-1)d OR 10=a1+4dAnda30=a1+(30-1)d OR 110=a1+29dNow use the 2 equations to solve for a1 & d. 10=a1+4d110=a1+29d (subtract the equations to cancel a1)-100= -25d So, d=4 and a1=-6 (now find the rule)an=a1+(n-1)dan=-6+(n-1)(4) OR an=-10+4n

Arithmetic SeriesThe sum of the terms in an arithmetic sequence

The formula to find the sum of a finite arithmetic series is:

# of terms1st TermLast Term

Example: Consider the arithmetic series 20+18+16+14+ .Find the sum of the 1st 25 terms.First find the rule for the nth term.an=22-2nSo, a25 = -28 (last term)Find n such that Sn=-760

-1520=n(20+22-2n)-1520=-2n2+42n2n2-42n-1520=0n2-21n-760=0(n-40)(n+19)=0n=40 or n=-19Always choose the positive solution!

Geometric Sequences and Series

1, 2, 4, 8, 16 is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it.The multiplier from each term to the next is called the common ratio and is usually denoted by r.Geometric Sequences and SeriesA geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.

Finding the Common RatioIn a geometric sequence, the common ratio can be found by dividing any term by the term preceding it.The geometric sequence 2, 8, 32, 128, has common ratio r = 4 since

Geometric Sequences and Seriesnth Term of a Geometric Sequence In the geometric sequence with first term a1 and common ratio r, the nth term an, is

The indicated sum of the terms of a geometric sequence is called a geometric series. You can derive a formula for the partial sum of a geometric series by subtracting the product of Sn and r from Sn as shown.

Using the Formula for the nth Term Example Find a5 and an for the geometric sequence 4, 12, 36, 108 ,

Solution Here a1= 4 and r = 36/ 12 = 3. Usingn=5 in the formula

In general

Modeling a Population of Fruit Flies Example A population of fruit flies grows in such a way that each generation is 1.5 times the previous generation. There were 100 insects in the first generation. How many are in the fourth generation.

Solution The populations form a geometric sequence with a1= 100 and r = 1.5 . Using n=4 in the formula for an gives

or about 338 insects in the fourth generation.

Geometric SeriesA geometric series is the sum of the terms of a geometric sequence .

In the fruit fly population model with a1 = 100 and r = 1.5, the total population after four generations is a geometric series:

Geometric Sequences and SeriesSum of the First n Terms of an Geometric Sequence If a geometric sequence has first term a1 and common ratio r, then the sum of the first n terms is given by where .

Finding the Sum of the First n Terms Example Find

Solution This is the sum of the first six terms of a geometric series with and r = 3. From the formula for Sn ,

.

Infinite Geometric Series If a1, a2, a3, is a geometric sequence and the sequence of sums S1, S2, S3, is a convergent sequence, converging to a number S. Then S is said to be the sum of the infinite geometric series

An Infinite Geometric Series Given the infinite geometric sequence

the sequence of sums is S1 = 2, S2 = 3, S3 = 3.5,

The calculator screen shows more sums, approaching a value of 4. So

Infinite Geometric SeriesSum of the Terms of an Infinite Geometric Sequence The sum of the terms of an infinite geometric sequence with first term a1 and common ratio r, where 1 < r < 1 is given by .

Finding Sums of the Terms of Infinite Geometric Sequences

Example Find

Solution Here and so

.

Fourier Series

In mathematics, infinite series are very important. They are used extensively in calculators and computers for evaluating values of many functions.The Fourier Series is really interesting, as it uses many of the mathematical techniques that you have learned before, like graphs, integration, differentiation, summation notation, trigonometry, etc

Infinite Series - Numbers A geometric progression is a set of numbers with a common ratio.Example: 1, 2, 4, 8, 16A series is the sum of a sequence of numbers.Example: 1 + 2 + 4 + 8 + 16 An infinite series that converges to a particular value has a common ratio less than 1. Example: 1 + 1/3 + 1/9 + 1/27 + ... = 3/2 When we expand functions in terms of some infinite series, the series will converge to the function as we take more and more terms.

Infinite Series Expansions of Functions

We learned before in the Infinite Series Expansions how to re-express many functions (like sin x, log x, ex, etc) as a polynomial with an infinite number of terms. We saw how our polynomial was a good approximation near some value x = a (in the case of Taylor Series) or x = 0 (in the case of Maclaurin Series). To get a better approximation, we needed to add more terms of the polynomial.

Fourier Series - A Trigonometric Infinite Series

In this lecture chapter we are also going to re-express functions in terms of an infinite series. However, instead of using a polynomial for our infinite series, we are going to use the sum of sine and cosine functions. Fourier Series is used in the analysis of signals in electronics. For example, the Fast Fourier Transform, which talks about pulse code modulation which is used when recording digital music.

Example

We will see functions like the following, which approximates a saw-tooth signal:

How does it work? As we add more terms to the series, we find that it converges to a particular shape.

Taking one extra term in the series each time and drawing separate graphs, we have:

f(t) = 1 (first term of the series):

Full Range Fourier Serieswhere an and bn are the Fourier coefficients, and is the mean value, sometimes referred to as the dc level.

Fourier Coefficients For Full Range Series Over Any Range -L TO L

Dirichlet Conditions Any periodic waveform of period p = 2L, can be expressed in a Fourier series provided that(a) it has a finite number of discontinuities within the period 2L;(b) it has a finite average value in the period 2L;(c) it has a finite number of positive and negative maxima and minima.When these conditions, called the Dirichlet conditions, are satisfied, the Fourier series for the function f(t) exists.Each of the examples in this chapter obey the Dirichlet Conditions and so the Fourier Series exists.

AL-HAMDULILLAH

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