Infinite sequence and series
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A study of one of the most interesting topics of mathematics or calculus. Sequences and Series.
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Infinite sequence and series
- 1. Shantilal Shah Engineering College, Bhavnagar. Mechanical Engineering Department
- 4. Sequences A sequence can be thought as a list of numbers written in a definite order ,,,,,, 4321 naaaaa { }na
- 5. Examples ( ) ( ) ( ) ( ) { } ,3,,3,2,1,03 , 3 11 ,, 27 4 , 9 3 , 3 2 3 11 , 1 ,, 4 3 , 3 2 , 2 1 1 3 −=− +− −−= +− + = + ∞ = nn nn n n n n n n n n n
- 6. Limit of a sequence (Definition) A sequence has the limit if for every there is a corresponding integer N such that We write { }na L ∞→→= ∞→ nasLaorLa nn n lim 0>ε NnwheneverLan ><− ,ε
- 7. Convergence/Divergence If exists we say that the sequence converges. Note that for the sequence to converge, the limit must be finite If the sequence does not converge we will say that it diverges Note that a sequence diverges if it approaches to infinity or if the sequence does not approach to anything n n a ∞→ lim
- 8. Divergence to infinity means that for every positive number M there is an integer N such that means that for every positive number M there is an integer N such that ∞= ∞→ n n alim NnwheneverMan >> , −∞= ∞→ n n alim NnwheneverMan >−< ,
- 9. The limit laws If and are convergent sequences and c is a constant, then { }na { }nb ( ) ( ) ccacac baba n n n n n n n n n nn n =⋅=⋅ ±=± ∞→∞→∞→ ∞→∞→∞→ lim,limlim limlimlim
- 10. The limit laws ( ) ( ) ( ) ( ) ( ) 00,limlim 0lim, lim lim lim limlimlim >>= ≠= ⋅=⋅ ∞→∞→ ∞→ ∞→ ∞→ ∞→ ∞→∞→∞→ n p n n p n n n n n n n n n n n n n n n nn n aandpifaa bif b a b a baba
- 11. Infinite Series Is the summation of all elements in a sequence. Remember the difference: Sequence is a collection of numbers, a Series is its summation. +++++=∑ ∞ = n n n aaaaa 321 1
- 12. Visual proof of convergence It seems difficult to understand how it is possible that a sum of infinite numbers could be finite. Let’s see an example ++++++= ++++++= ∑ ∑ ∞ = ∞ = n n n n n n 2 1 16 1 8 1 4 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 432 1
- 13. Convergence/Divergence We say that an infinite series converges if the sum is finite, otherwise we will say that it diverges. To define properly the concepts of convergence and divergence, we need to introduce the concept of partial sum N N n nN aaaaaS ++++== ∑= 321 1
- 14. Convergence/Divergence The partial sum is the finite sum of the first terms. converges to if and we write: If the sequence of partial sums diverges, we say that diverges. th N NS ∑ ∞ =1n na S SSN N = ∞→ lim ∑ ∞ = = 1n naS ∑ ∞ =1n na
- 15. Laws of Series If and both converge, then Note that the laws do not apply to multiplication, division nor exponentiation. ∑ ∞ =1n na ∑ ∞ =1n nb ( ) ∑∑ ∑∑∑ ∞ = ∞ = ∞ = ∞ = ∞ = ⋅=⋅ ±=± 11 111 n n n n n nn n n n n acac baba
- 16. Divergence Test If does not converge to zero, then diverges. Note that in many cases we will have sequences that converge to zero but its sum diverges { }na ∑ ∞ =1n na ( ) ∑∑∑∑∑ ∞ = ∞ = ∞ = ∞ = ∞ = − 11 2 111 sin 111 1 nnnnn n n nnn
- 17. Proof Divergence Test If , then ( ) 1 1 1321 1321 − − − − −=⇒ += +++++= +++++= nnn nnn nnn nnn SSa aSS aaaaaS aaaaaS ∑ ∞ = = 1n n Sa
- 18. Geometric Series +⋅+⋅+⋅+⋅+=⋅∑ ∞ = 432 0 rcrcrcrccrc n n Note that in this case we start counting from zero. Technically it doesn’t matter, but we have to be careful because the formula we will use starts always at n=0. First term multiplied by r Second term multiplied by r Third term multiplied by r
- 19. Geometric Series If we multiply both sides by r we get If we subtract (2) from (1), we get )1(32 0 N N N n n N rcrcrcrccS rcS ⋅++⋅+⋅+⋅+= ⋅= ∑= )2(1432 + ⋅++⋅+⋅+⋅+⋅=⋅ N N rcrcrcrcrcSr ( ) ( ) ( ) r rc S rcrS rccSrS N NN N N NN − − =⇒ −=− ⋅−=⋅− + + + 1 1 11 1 1 1
- 20. Geometric Series An infinite GS diverges if , otherwise1≥r 1, 1 1 1, 1 1, 10 < − =⋅ < − ⋅ =⋅ < − =⋅ ∑ ∑ ∑ ∞ = ∞ = ∞ = r r term rc r r rc rc r r c rc st Mn n M Mn n n n
- 21. Examples ( ) ( ) ( ) ( ) ∑∑ ∑∑∑ ∑∑∑∑ ∞ = ∞ = ∞ = ∞ = − ∞ = ∞ = ∞ = − ∞ = ∞ = + + − −⋅ 10 11 1 1 2000 52 ln 6 23 26.0 5 1 3 1 2113 nn n nn n n n n n n n n n n n n n n n n
- 22. P-Series A p-series is a series of the form Convergence of p-series: ++++=∑ ∞ = ppp n p n 4 1 3 1 2 1 1 11 1 ≤ > =∑ ∞ = 1 11 1 pforDiverges pforConverges nn p
- 24. Comparison Test Assume that there exists such that for 1. If converges, then also converges. 2. If diverges, then also diverges. if diverges this test does not help Also, if converges this test does not help 0>M nn ba ≤≤0 Mn ≥ ∑ ∞ =1n nb ∑ ∞ =1n na ∑ ∞ =1n na ∑ ∞ =1n nb ∑ ∞ =1n nb ∑ ∞ =1n na
- 25. Limit Comparison Test Let and be positive sequences. Assume that the following limit exists If , then converges if and only if converges. (Note that L can not be infinity) If and converges, then converges { }na { }nb n n n b a L ∞→ = lim 0>L ∑ ∞ =1n na ∑ ∞ =1n nb0=L ∑ ∞ =1n nb ∑ ∞ =1n na
- 27. Absolute/Conditional Convergence is called absolutely convergent if converges Absolute convergence theorem: If convs. Also convs. (In words) if convs. Abs. convs. ∑ ∞ =1n na ∑ ∞ =1n na ∑ ∞ =1n na ⇒∑ ∞ =1n na ∑ ∞ =1n na ⇒∑ ∞ =1n na ( ) ∑∑ ∞ = ∞ = − − 1 2 1 1 2 1 n n n n n
- 28. Ratio Test Let be a sequence and assume that the following limit exists: If , then converges absolutely If , then diverges If , the Ratio Test is INCONCLUSIVE { }na n n n a a 1 lim + ∞→ =ρ 1<ρ ∑ ∞ =1n na 1=ρ 1>ρ ∑ ∞ =1n na ( ) ∑∑∑∑∑ ∞ = − ∞ = ∞ = ∞ = ∞ = − 1 2 1 2 11 2 1 100 ! 1 2! 1 nnn n n n n n nn nn n Examples
- 29. Root Test Let be a sequence and assume that the following limit exists: If , then converges absolutely If , then diverges If , the Ratio Test is INCONCLUSIVE { }na n n n aL ∞→ = lim 1<L ∑ ∞ =1n na 1=L 1>L ∑ ∞ =1n na ∑∑∑∑ ∞ = − ∞ = ∞ = ∞ = + 1 2 1 2 1 2 1 232 nnn n n n nn n n n Examples
- 30. Power Series A power series is a series of the form: ( ) ( ) ( ) +−+−+=− +++++= ∑ ∑ ∞ = ∞ = 2 21 0 0 2 21 0 0 axcaxccaxc xcxcxccxc n n n n n n n n
- 31. Power Series Theorem: For a given power series there are 3 possibilities: 1. The series converges only when 2.The series converges for all 3. There is a positive number R, such that the series converges if and diverges if ( )∑ ∞ = − 0n n n axc ax = x Rax <− Rax >−
- 32. Taylor & Maclaurin Series Let , then therefore , ,234)0(,23)0(,2)0(,)0(,)0( 4 )( 3210 afafafafaf IV ⋅⋅=⋅=′′′=′′=′= +++++== ∑ ∞ = 4 4 3 3 2 210 0 )( xaxaxaxaaxaxf n n n n n nk k x n f xf k f a ∑ ∞ = =⇒= 0 )()( ! )0( )( ! )0( 2 2 cossin xxx eexxxe Examples −