Question

In: Advanced Math

Sn = (1+(1/n))^n (a) Prove Sn is strictly increasing (b) bounded below by 2 and above...

Sn = (1+(1/n))^n

(a) Prove Sn is strictly increasing (b) bounded below by 2 and above by 3

(c) Sn converges to e

(d) Obtain an expression for e

(e) Prove e is irrational

Solutions

Expert Solution


Related Solutions

Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>
Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>
Prove that if a sequence is bounded, then limsup sn is a real number.
Prove that if a sequence is bounded, then limsup sn is a real number.
1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove...
1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove that if A is a bounded set, then A-bar is compact.
Show that if S is bounded above and below, then there exists a number N >...
Show that if S is bounded above and below, then there exists a number N > 0 for which - N < or equal to x < or equal to N if x is in S
4. Prove that for any n∈Z+, An ≤Sn.
4. Prove that for any n∈Z+, An ≤Sn.
Prove that if f is a bounded function on a bounded interval [a,b] and f is...
Prove that if f is a bounded function on a bounded interval [a,b] and f is continuous except at finitely many points in [a,b], then f is integrable on [a,b]. Hint: Use interval additivity, and an induction argument on the number of discontinuities.
Prove that Sn is generated by {(i i + 1) I 1 < i < n - 1}.
  Prove that Sn is generated by {(i i + 1) I 1 < i < n - 1}.  
(a) Prove that Sn is generated by the elements in the set {(i i+1) : 1≤i≤n}....
(a) Prove that Sn is generated by the elements in the set {(i i+1) : 1≤i≤n}. [Hint: Consider conjugates, for example (2 3) (1 2) (2 3)−1.] (b) ProvethatSn isgeneratedbythetwoelements(12)and(123...n) for n ≥ 3. (c) Prove that H = 〈(1 3), (1 2 3 4)〉 is a proper subgroup of S4.
Ex 4. (a) Prove by induction that ∀n∈N,13+ 23+ 33+···+n3=[(n(n+ 1))/2]2 b) Prove by induction that...
Ex 4. (a) Prove by induction that ∀n∈N,13+ 23+ 33+···+n3=[(n(n+ 1))/2]2 b) Prove by induction that 2n>2n for every natural number n≥3.
1: (an) and (bn) are bounded sequences: (a) prove that limsup(-an) = -liminf(an) (b) for any...
1: (an) and (bn) are bounded sequences: (a) prove that limsup(-an) = -liminf(an) (b) for any c>0, prove that limsup(can) = climsup(an) and liminf(can) = climinf(an) (c) prove that limsup(an+bn) ≤ (limsup(an)) + (limsup(bn)) and liminf(an+bn) ≥ (liminf(an)) + (liminf(bn)) (d) If an and bn are made of nonnegative terms, prove that limsup(anbn) ≤ (limsup(an)) x (limsup(bn)) and liminf(anbn) ≥ (liminnf(an)) x (liminf(bn)) (e) prove that limsup(an+1) = limsup(an) and liminf(an+1) = liminf(an)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT