If #a_n# converges and #lim_(n->oo) a_n -b_n=c#, where c is a constant, does #b_n# converge?

Answer 1

#lim_(n->oo) b_n = -c+ lim_(n->oo) a_n#

Let:

#lim_(n->oo) a_n = L#
Then for any number #epsilon > 0# we can find #N_epsilon# such that:
#n > N_epsilon => abs (a_n-L) < epsilon/2#
Similarly, as #lim_(n->oo) (a_n-b_n) = c# for the same #epsilon# we can find #M_epsilon# such that:
#n > M_epsilon => abs(a_n-b_n -c) < epsilon/2#

Consider now the quantity:

#abs (-b_n +L - c) = abs(a_n-b_n -c-a_n+L)#

using the triangular inequality:

#abs (-b_n +L - c) <= abs(a_n-b_n -c)+abs(-a_n+L)#
But then if we take #P_epsilon = max(M_epsilon,N_epsilon)# we have:
#n > P_epsilon => abs (-b_n +L - c) < epsilon/2+epsilon/2#

which means:

#lim_(n->oo) b_n = L-c#
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Answer 2

If ( a_n ) converges and ( \lim_{n \to \infty} (a_n - b_n) = c ), where ( c ) is a constant, then ( b_n ) also converges, and its limit is ( \lim_{n \to \infty} b_n = \lim_{n \to \infty} (a_n - c) ).

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Answer 3

If (a_n) converges and (\lim_{n \to \infty} (a_n - b_n) = c), where (c) is a constant, then (b_n) converges as well.

We know that if (a_n) converges, then (\lim_{n \to \infty} a_n = L) for some finite limit (L).

Given (\lim_{n \to \infty} (a_n - b_n) = c), we can rewrite it as (\lim_{n \to \infty} a_n - \lim_{n \to \infty} b_n = c).

Since (\lim_{n \to \infty} a_n = L) (as (a_n) converges), we have (L - \lim_{n \to \infty} b_n = c).

Rearranging, we get (\lim_{n \to \infty} b_n = L - c), which shows that (b_n) converges to (L - c), a finite limit.

Therefore, if (a_n) converges and (\lim_{n \to \infty} (a_n - b_n) = c), then (b_n) converges.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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