# barcode in ssrs report CAUCHY S CONVERGENCE CRITERION 2.24. Prove Cauchy s convergence criterion as stated on Page 25. in .NET Painting QR in .NET CAUCHY S CONVERGENCE CRITERION 2.24. Prove Cauchy s convergence criterion as stated on Page 25.

CAUCHY S CONVERGENCE CRITERION 2.24. Prove Cauchy s convergence criterion as stated on Page 25.
Denso QR Bar Code Decoder In Visual Studio .NET
Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications.
QR Code ISO/IEC18004 Maker In .NET
Using Barcode encoder for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications.
Necessity. Suppose the sequence fun g converges to l. Then given any  > 0, we can nd N such that jup lj < =2 for all p > N Then for both p > N and q > N, we have jup uq j j up l l uq j @ jup lj jl uq j < =2 =2  Su ciency. Suppose jup uq j <  for all p; q > N and any  > 0. Then all the numbers uN ; uN 1 ; . . . lie in a nite interval, i.e., the set is bounded and in nite. Hence, by the Bolzano Weierstrass theorem there is at least one limit point, say a. If a is the only limit point, we have the desired proof and lim un a. Suppose there are two distinct limit points, say a and b, and suppose b > a (see Fig. 2-1). By de nition of limit points, we have jup aj < b a =3 for infinnitely many values of p juq bj < b a =3 for infinitely many values of q Then since b a b uq uq up up a , we have jb aj b a @ jb uq j jup uq j jup aj 3 1 2
Using Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications.
Encoding Bar Code In Visual Studio .NET
Using Barcode creation for .NET Control to generate, create bar code image in .NET framework applications.
b_a 3 a b_a 3 b
Scanning Bar Code In VS .NET
Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications.
QR Code Maker In Visual C#
Using Barcode creation for .NET Control to generate, create Quick Response Code image in .NET framework applications.
juq lj < =2 for all q > N
Quick Response Code Maker In .NET
Using Barcode drawer for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications.
Making QR Code In VB.NET
Using Barcode maker for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications.
Fig. 2-1
Create EAN13 In VS .NET
Using Barcode maker for VS .NET Control to generate, create EAN-13 image in VS .NET applications.
Encoding ANSI/AIM Code 39 In VS .NET
Using Barcode generation for .NET Control to generate, create USS Code 39 image in VS .NET applications.
Using (1) and (2) in (3), we see that jup uq j > b a =3 for in nitely many values of p and q, thus contradicting the hypothesis that jup uq j <  for p; q > N and any  > 0. Hence, there is only one limit point and the theorem is proved.
Drawing Barcode In .NET
Using Barcode generator for VS .NET Control to generate, create barcode image in Visual Studio .NET applications.
Print 2 Of 5 Interleaved In .NET Framework
Using Barcode generation for .NET framework Control to generate, create USS ITF 2/5 image in VS .NET applications.
INFINITE SERIES 2.25. Prove that the in nite series (sometimes called the geometric series) a ar ar2 (a) converges to a= 1 r if jrj < 1,
Bar Code Decoder In Visual Basic .NET
Using Barcode decoder for .NET framework Control to read, scan read, scan image in VS .NET applications.
Painting ANSI/AIM Code 128 In Visual Basic .NET
Using Barcode generation for .NET Control to generate, create Code 128C image in .NET framework applications.
Let Then Subtract,
Bar Code Generator In Objective-C
Using Barcode maker for iPhone Control to generate, create barcode image in iPhone applications.
Barcode Encoder In None
Using Barcode drawer for Word Control to generate, create bar code image in Office Word applications.
1 X n 1
Generate UCC - 12 In None
Using Barcode encoder for Online Control to generate, create UPC Code image in Online applications.
Bar Code Maker In Visual C#
Using Barcode creation for VS .NET Control to generate, create barcode image in Visual Studio .NET applications.
arn 1
Code39 Printer In Objective-C
Using Barcode generator for iPhone Control to generate, create Code39 image in iPhone applications.
USS Code 39 Printer In VB.NET
Using Barcode maker for .NET Control to generate, create Code 3 of 9 image in .NET applications.
(b) diverges if jrj A 1.
ar ar2 arn 1 arn arn
Sn a ar ar2 arn 1 rSn 1 r Sn a
SEQUENCES
[CHAP. 2
Sn If jrj < 1; lim Sn lim
n!1 n!1
a 1 rn 1 r
a 1 rn a by Problem 7: n!1 1 r 1 r
(b) If jrj > 1, lim Sn does not exist (see Problem 44).
2.26. Prove that if a series converges, its nth term must necessarily approach zero.
Since Sn u1 u2 un , Sn 1 u1 u2 un 1 we have un Sn Sn 1 . If the series converges to S, then
lim un lim Sn Sn 1 lim Sn lim Sn 1 S S 0
n!1 n!1 n!1
2.27. Prove that the series 1 1 1 1 1 1
1 X 1 n 1 diverges. n 1
Method 1: lim 1 n 6 0, in fact it doesn t exist. Then by Problem 2.26 the series cannot converge, i.e., it diverges.
Method 2: The sequence of partial sums is 1; 1 1; 1 1 1; 1 1 1 1; . . . i.e., 1; 0; 1; 0; 1; 0; 1; . . . . Since this sequence has no limit, the series diverges.
MISCELLANEOUS PROBLEMS 2.28. If lim un l, prove that lim
n!1 n!1
u 1 u2 un l. n
Let un vn l.
v1 v2 vn 0 if lim vn 0. Now n!1 n v1 v2 vn v1 v2 vP vP 1 vp 2 vn n n n We must show that lim
so that
  v 1 v 2 v n    @ jv1 v2 vP j jvP 1 j jvP 2 j jvn j   n n n Then
Since lim vn 0, we can choose P so that jvn j < =2 for n > P.
jvP 1 j jvP 2 j jvn j =2 =2 =2 n P =2  < < n n n 2 After choosing P we can choose N so that for n > N > P, jv1 v2 vP j  < n 2 Then using (2) and (3), (1) becomes   v 1 v 2 v n     <    2 2 n thus proving the required result.
for n > N
2.29. Prove that lim 1 n n2 1=n 1.
Let 1 n n2 1=n 1 un where un A 0.
Now by the binomial theorem,
CHAP. 2]
SEQUENCES
1 n n2 1 un n 1 nun
n n 1 2 n n 1 n 2 3 un un un n 2! 3!
Then 1 n n2 > 1 Hence, lim u3 0 and n
n n 1 n 2 3 6 n2 n un or 0 < u3 < : n 3! n n 1 n 2 lim un 0: Thus lim 1 n n2 1=n lim 1 un 1: