2. Let an = 5n 4n- 3 This sequence converges to L = . (a) Find the smallest N EN such that an - L < 10-4 for all n ≥ N. (b) Show that, for all n > 12, |an - L < 1/n. Deduce an integer M (not necessarily the smallest!) such that an - L < 10-6 for all n ≥ M. (c) Prove from first principles that an converges to . 5
2. Let an = 5n 4n- 3 This sequence converges to L = . (a) Find the smallest N EN such that an - L < 10-4 for all n ≥ N. (b) Show that, for all n > 12, |an - L < 1/n. Deduce an integer M (not necessarily the smallest!) such that an - L < 10-6 for all n ≥ M. (c) Prove from first principles that an converges to . 5
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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![2. Let an
This sequence converges to L = 3.
(a) Find the smallest N E N such that |an − L| < 10−4 for all n ≥ N.
(b) Show that, for all n > 12, |an - L| < 1/n. Deduce an integer M (not necessarily
the smallest!) such that |an - L| < 10−6 for all n ≥ M.
5n
4n - 3
=
5
(c) Prove from first principles that an converges to .](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F0daea853-7471-4e1f-9c4b-60fdd772c326%2Fcrxrqd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let an
This sequence converges to L = 3.
(a) Find the smallest N E N such that |an − L| < 10−4 for all n ≥ N.
(b) Show that, for all n > 12, |an - L| < 1/n. Deduce an integer M (not necessarily
the smallest!) such that |an - L| < 10−6 for all n ≥ M.
5n
4n - 3
=
5
(c) Prove from first principles that an converges to .
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