Real Analysis Answers

Give an example of a convergent series 􏰄 an such that 􏰄 a2n is not convergent

If f(x):= x² for xe [0, 4), calculate the following Riemann sums, where Pi has the same partition points as in Exercise 1, and the tags are selected as indicated. (a) P, with the lags at the left endpoints of the subintervals. (b) Pi with the tags at the right endpoints of the subintervals. (c) P2 with the tags at the left endpoints of the subintervals. (d) P2 with the tags at the right endpoints of the subintervals.

1) If ((xn,yn)) is a bounded sequence, then

((xn,yn)) has convergent subsequence.

2)((xn,yn)) is Cauchy if and only if ((xn,yn)) is convergent.

1) If ((xn,yn)) is a bounded sequence, then

((xn,yn)) has convergent subsequence.

2)((xn,yn)) is Cauchy if and only if ((xn,yn)) is convergent.

(Cauchy criterion for limits of functions) Let D be a subset of R2 and (x0,y0) € R2 be such that D contains r nhbd of (x0, y0) \ {(x0, y0)} for some r >0 and let f: D to R be a function. Then the limit of f(x,y) as (x, y) tends to (x0, y0) exists if for every epsilon > 0, there is delta >0 such that (x,y),(u,v) € D intersection delta nhbd of (x0, y0) \{(x0, y0)} implies |f(x,y) - f(u,v)|< epsilon.

Let D be a subset of R2 and (x0,y0) € R2 be such that D contains r nhbd of (x0, y0) \ {(x0, y0)} for some r >0 and let f: D to R be a function. Then the limit of f(x,y) as (x, y) tends to (x0, y0) exists if and only if there is l € R satisfying the following epsilon delta condition: For every epsilon > 0, there is delta >0 such that (x,y) € D intersection delta nhbd of (x0, y0) and (x, y) not equal to (x0,y0) implies |f(x,y) - l|< epsilon.

Let D be a subset of R2 and (x0,y0) € R2 be such that D contains delta nhbd of (x0, y0) \ {(x0, y0)} for some r >0 and let f: D-R be a function. Thent thelimit of f(x,y) as (x, y) tends to (x0, y0) exists if and only if there is l € R satisfying the following epsilon delta condition: For every epsilon > 0, there is delta >0 such that (x,y) € D intersection delta nhbd of (x0, y0) and (x, y)  (20,40) is not equal to |f(x,y) - l|< epsilon.

1) If ((xn,yn)) is a bounded sequence, then

((xn,yn)) has convergent subsequence.

2)((xn,yn)) is Cauchy if and only if ((xn,yn)) is convergent.