Question

2. Sequence Convergence¶ (To be answered in LaTex)

Show that the following sequence limits using either an ?−?ε−δ argument

a) lim (3?+1) / (2?+5) = 3/2 answer in LaTex

b) lim (2?) / (n+2) =2 answer in LaTex

c) lim [(1/?) − 1/(?+1)] = 0 answer in LaTex

Answer #1

(a) Let $\epsilon >0$, then by Archimedian property there exists $N \in \mathbf{N}$ such that for all $n > N$ we have $\frac{1}{2n+5} < \frac{2 \epsilon}{5}$.

Now $|\frac{3n+1}{2n+5}- \frac{3}{2}| = |\frac{13}{2(2n+5)}| < \epsilon$ for all $n > N$. Hence $\lim \frac{3n+1}{2n+5} = \frac{3}{2}$.

(b) Let $\epsilon >0$, then by Archimedian property there exists $N \in \mathbf{N}$ such that for all $n > N$ we have $\frac{1}{n+2} < \frac{ \epsilon}{4}$.

Now $|\frac{2n}{n+2}- 2| = |\frac{4}{n+2}| < \epsilon$ for all $n > N$. Hence $\lim \frac{2n}{n+2} = 2$.

(c) Let $\epsilon >0$, then by Archimedian property there exists $N \in \mathbf{N}$ such that for all $n > N$ we have $\frac{1}{n} < \frac{ \epsilon}{2}$.

Now $|\frac{1}{n}- \frac{1}{n+1}| < |\frac{1}{n}|+ |\frac{1}{n+1}| < \epsilon$ for all $n > N$. Hence $\lim \frac{1}{n}- \frac{1}{n+1} = 0$.

Using the definition of convergence of a sequence, prove that the
sequence converges to the proposed limit.
lim (as n goes to infinity) 1/(n^2) = 0

hi guys , using this definition for limits in higher dimensions
:
lim (x,y)→(a,b) f(x, y) = L
if 1. ∃r > 0 s.th. f(x, y) is defined when 0 < || (x, y) −
(a, b) || < r
and 2. given ε > 0 we can find δ > 0 s.th. 0 < || (x,
y) − (a, b) || < δ =⇒ | f(x, y) − L | < ε
how do i show that this is...

Use the Monotone Convergence Theorem to show that each
sequence
converges.
a)an= -(2/3)^n
b)an= 1+ 1/n
c) 2/(-n)^2

1) Let c ∈ R. Discuss the convergence of the sequence
an = cn
2) Suppose that the sequence {an} converges to l and
that an > 0 for all n. Show that l ≥ 0

Show that the sequence {1- (-1)n} has no
subsequential limits besides 0 and 2

Determine the limits of the following sequences. The prove your
claims using an e - N argument.
a. an = n / (n2 + 1)
b. bn = (4n + 3) / (7n - 5)
c. cn = 1/n sin(n)

Use the Maclaurin series for cos(?) , ????(?), and ??
to evaluate the following limits:
a. lim ?→0 −? − 1+?x / 5? 2 .
b. lim ?→0 ? − ???? / ?3 ????

Compute the following limits and show all your work:
(a) A = lim x→∞ [3x^2 −4/x^2 +10lnx]
(b) B = lim x→2+ [(x-2)^2ln(x-1)]
̧
.

please show all work Evaluate each of the following limits, for
after lim the part with x-> and then a number is below the lim
and then after is the fraction part
1) lim x->3 (x^2-2x-3/x^2-5x+6)
2) limx->2 (x-2/square root(2x)-2)
3) lim x->inf (3x^5-7x^3/-5x^5+x^3-9)

1. Determine the convergence or divergence of the sequence with
given ??h term
(a) an=4-5/(n^2+1)
(b) an= 1/√?
(c) an= (sin√?)/ √?

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