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INFINITE SEQUENCE AND SERIES

INFINITE SEQUENCE AND SERIES

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· Sequence:

· A sequence is a set of numbers in a specific order.

· For example:

·

·

·

· Bounded and Unbounded Sequence:

· A sequence is said to be bounded above if there exist a real number K such that for every. K is called an upper bound for.

· A sequence is said to be bounded below if there exist a real number K such that for every. K is called a lower bound for.

· A sequence is said to be bounded if it is both bounded above and bounded below.

· Monotonic Sequence:

· A sequence is said to be an increasing sequence if , for every

· A sequence is said to be an decreasing sequence if , for every

· A sequence is said to be a monotonic if it is either increasing or decreasing.

· Convergent Sequence and Divergent Sequence:

· A sequence is said to be convergent if

Where is the nth term of given sequence and ‘l’ is any finite real number.

· A sequence is said to be divergent if

Where is the term of given sequence.

· Oscillating Sequence:

· A sequence which is neither convergent nor divergent is said to be an oscillating sequence.

· Result on Convergent or Divergent of Sequence:

· An increasing sequence which is bounded above is always convergent.

· A decreasing sequence which is bounded below is always convergent.

· Every bounded and monotonic sequence is convergent.

· The Sandwich Theorem:

· Let {}, {}, {} be the sequence of real numbers.

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CHARACTERISTIC OF SEQUENCE AND LIMIT

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Find the following limits:

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CONVERGENCE OF SEQUENCE

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Diverges to ,

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.

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3

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June 2012

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· Positive Term Series:

· If all terms after few negative terms in an infinite series are positive, such a series is a positive terms series.

· Test for Geometric Series:-

·

Where ‘r’ is common ratio and ‘a’ is the term, then given series is

·

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GEOMETRIC SERIES

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,

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June2015

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Define the Geometric series and find the sum of .

4/5

June2015

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June 2014

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· Zero Test (Cauchy’s Fundamental Test or nth Term Test):-

· Suppose is given series where is the nth term of the series.

Ifor does not exist then given series is divergent.

NOTE: This test is only for divergent series.

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ZERO TEST/CAUCHY’S FUNDAMENTAL TEST

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· Procedure of checking Convergent or Divergent using:-

Find nth term of given series if it is not directly given.

Using partial fraction separate it into two or three individual term.

Find Sn i.e. sum of first n term using above individual term.

If above limit is finite then series is convergent otherwise divergent.

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TELESCOPIC SERIES

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June 2014

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Dec 2011

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· Integral Test:-

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INTEGRAL TEST

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Convergent

June 2013

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Convergent

Dec 2013

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Both series are

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Show that p-series (p is real constant) converges if and diverges if

June2015

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Divergent, , convergent, convergent, convergent

· Direct Comparison Test:-

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Hint:

· If bigger series is convergent then smaller series is also convergent.

· If smaller series is divergent then bigger series is also divergent.

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DIRECT COMPARISION TEST (DCT)

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Dec 2013

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· Limit Comparison Test:-

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LIMIT COMPARISION TEST (LCT)

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, Convergent, Convergent, Convergent

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, convergent, convergent, convergent

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June 2014

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June 2013

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Is the series converges or diverges?

Divergent

June2015

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Dec 2013

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June 2012

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June 2012

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.

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Convergent

Jan 2013

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All the series are Divergent

· D’ Alembert’s Ratio Test:-

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RATIO TEST

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All the series are

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Check convergence of

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Test for convergence or divergence:

Convergent

June2015

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Convergent

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June 2012

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June 2014

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June 2012

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· Rabbe’s Higher Ratio Test:-

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RABBE’S TEST

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June 2014

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Convergent

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Convergent

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Convergent

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Convergent

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Divergent

· Cauchy’s nth Root(Radical) Test:-

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ROOT TEST

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All the series are Divergent

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Convergent

Dec 2013

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June 2014

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Convergent

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Convergent

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June 2012

June 2014

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· Alternative Series :

· A series with alternate positive and negative terms is known as an alternative series.

· Leibnitz’s Test For Alternative Series:

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.

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LEIBNITZ TEST

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June 2014

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Both series are

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June 2014

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Dec 2011

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June 2013

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· Absolute Convergent Series:

· The given alternative series is said to be absolute convergent if

is convergent.

· Any absolute convergent series is always convergent.

· Conditional Convergent Series:

· The given alternative series is said to be conditional convergent if

· is convergent and is not divergent.

· Power Series :

· An infinite series of the form

is known as power series in standard form, where is the coefficient of the nth term, c is a constants and x varies around c.

· if c = 0 in above series then it is called power series in power of .

· [Theorem #] Convergence of power series:

· Let be a power series. Then exactly one of the following conditions hold.

1) The series always converges at.

2) The series converges for all x.

3) There is some positive number R such that series converges for and diverges for.

· Radius and Interval of Convergence For Power Series:

· An interval in which power series converges is called the interval of convergence and the half length of the interval of convergence is called the radius of convergence.

· If a power series is convergent for all values of x, then interval of convergence will be and the radius of convergence will be.

· In above theorem R is radius of convergence of the power series and it can be obtained from either of the formulas as follows:

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ABSULUTE CONVEGENCE

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following series:

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CONDITIONAL CONVERGENCE

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RADIUS AND INTERVAL OF CONVERGENCE

a) Find the radius of convergence.

b) Find the interval (range) of convergence.

c) For what values of x does the series converge absolutely.

d) For what values of x does the series converge conditionally.

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MIX EXAMPLES

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Find radius of convergence and interval of convergence of the series

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Dec 2011

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· Taylor’s Series { 1st form or power of }:

· If f(x) possesses derivatives of all order at point a then Taylor’s series of given function f(x) at point a is given by

· Taylor’s Series { 2nd form or }:

· If f(x) possesses derivatives of all order at point a then Taylor’s series of given function f(a+h) at point a is given by

· If we take x = a + h ( i.e. x – a = h ) in the 1st form of Taylor’s series then we get the 2nd form of Taylor’s series.

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TAYLOR’S SERIES { 1ST FORM, POWER OF (x-a) }

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June 2013

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June 2012

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TAYLOR’S SERIES { 2ND FORM, FORM }

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Dec 2013

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Dec 2013

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Define the Taylor’s for the function of one variable and using it show that

Where

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· Maclaurin’s Series :

· If f(x) possesses derivatives of all order at point 0 then Maclaurin’s series of given function f(x) at point 0 is given by

· If we take point a = 0 in the 1st form of Taylor’s series then we get the Maclaurin’s series.

· Binomial Series:

· Remember following Formulae:

· Expansion of the sine function(only odd power).

· Expansion of the cosine function(only even power).

· Expansion of the tangent function (only odd power).

· Expansion of the function (even & odd power).

· Expansion of the exponential function (even & odd power).

· Expansion of the logarithmic function (even & odd power).

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MACLAURIN’S SERIES

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June 2013

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June 2014

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INTEGRATION AND DIFFERENTIATION

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TRIGONOMETRIC SUBSTITUTIONS

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Dec 2013

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