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INFINITE SEQUENCE AND SERIES
INFINITE SEQUENCE AND SERIES
UNIT-1
· 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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Dec 2011
· 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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3
GEOMETRIC SERIES
1)
,
2)
June2015
3)
Define the Geometric series and find the sum of .
4/5
June2015
4)
June 2014
5)
· 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
1)
Convergent
June 2013
2)
Convergent
Dec 2013
3)
Both series are
4)
Show that p-series (p is real constant) converges if and diverges if
June2015
5)
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Divergent, , convergent, convergent, convergent
· Direct Comparison Test:-
·
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)
1)
, Convergent, Convergent, Convergent
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, convergent, convergent, convergent
3)
June 2014
4)
June 2013
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Is the series converges or diverges?
Divergent
June2015
6)
Dec 2013
7)
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
1)
All the series are
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Check convergence of
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Test for convergence or divergence:
Convergent
June2015
5)
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Convergent
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June 2012
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June 2014
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June 2012
June 2014
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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
1)
All the series are Divergent
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Convergent
Dec 2013
4)
5)
June 2014
6)
Convergent
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Convergent
8)
9)
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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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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Dec 2013
· 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:
13
ABSULUTE CONVEGENCE
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following series:
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CONDITIONAL CONVERGENCE
1)
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June 2013
4)
June 2013
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15
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.
1)
2)
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MIX EXAMPLES
1)
Find radius of convergence and interval of convergence of the series
2)
Find radius of convergence and interval of convergence of the series
Dec 2011
Dec 2013
3)
Find radius of convergence and interval of convergence of the series
4)
Find radius of convergence and interval of convergence of the series
5)
Find radius of convergence and interval of convergence of the series
Dec 2013
6)
Find radius of convergence and interval of convergence of the series
7)
Find radius of convergence and interval of convergence of the series
8)
Dec 2013
9)
Jan 2013
· 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) }
1)
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.
June 2013
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June 2014
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Dec 2011
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June 2012
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June 2014
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Jan 2013
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TAYLOR’S SERIES { 2ND FORM, FORM }
1)
Dec 2013
2)
Dec 2011
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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
June2015
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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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Dec 2013
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June 2014
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June 2012
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Dec 2011
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.
20
INTEGRATION AND DIFFERENTIATION
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TRIGONOMETRIC SUBSTITUTIONS
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Dec 2013
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