Studdy AI Logo
Math  /  Calculus

Question4 Mark for Review - -1 - - - -
If the infinite series ∑n=0∞an\sum_{n=0}^{\infty} a_{n} diverges, Sn=∑k=0nakS_{n}=\sum_{k=0}^{n} a_{k}, and lim⁡n→∞bn≠0\lim _{n \rightarrow \infty} b_{n} \neq 0, which of the following statements must be true?
1. lim⁡n→∞an≠0\lim _{n \rightarrow \infty} a_{n} \neq 0
11. lim⁡n→∞Sn\lim _{n \rightarrow \infty} S_{n} does not exist. II. ∑n=0∞bn\sum_{n=0}^{\infty} b_{n} diverges. (A) Ionly (B) Il only (C) II and ili only
D I and III only

Studdy Solution

STEP 1

1. The series ∑n=0∞an\sum_{n=0}^{\infty} a_{n} diverges.
2. Sn=∑k=0nakS_{n} = \sum_{k=0}^{n} a_{k} is the partial sum of the series ∑n=0∞an\sum_{n=0}^{\infty} a_{n}.
3. lim⁡n→∞bn≠0\lim _{n \rightarrow \infty} b_{n} \neq 0.
4. We need to determine which statements must be true based on the given conditions.

STEP 2

1. Analyze statement I: lim⁡n→∞an≠0\lim _{n \rightarrow \infty} a_{n} \neq 0.
2. Analyze statement II: lim⁡n→∞Sn\lim _{n \rightarrow \infty} S_{n} does not exist.
3. Analyze statement III: ∑n=0∞bn\sum_{n=0}^{\infty} b_{n} diverges.
4. Determine which combination of statements is true.

STEP 3

Analyze statement I: lim⁡n→∞an≠0\lim _{n \rightarrow \infty} a_{n} \neq 0.
For a series ∑n=0∞an\sum_{n=0}^{\infty} a_{n} to converge, a necessary condition is that lim⁡n→∞an=0\lim_{n \to \infty} a_{n} = 0. Since the series diverges, it is possible that lim⁡n→∞an≠0\lim_{n \to \infty} a_{n} \neq 0, but this is not a must. The series could diverge even if lim⁡n→∞an=0\lim_{n \to \infty} a_{n} = 0 (e.g., the harmonic series).

STEP 4

Analyze statement II: lim⁡n→∞Sn\lim _{n \rightarrow \infty} S_{n} does not exist.
Since ∑n=0∞an\sum_{n=0}^{\infty} a_{n} diverges, the sequence of partial sums SnS_{n} does not converge to a finite limit. Therefore, lim⁡n→∞Sn\lim _{n \rightarrow \infty} S_{n} does not exist.

STEP 5

Analyze statement III: ∑n=0∞bn\sum_{n=0}^{\infty} b_{n} diverges.
Given that lim⁡n→∞bn≠0\lim _{n \rightarrow \infty} b_{n} \neq 0, the series ∑n=0∞bn\sum_{n=0}^{\infty} b_{n} must diverge. For a series to converge, the terms must approach zero, which is not the case here.

STEP 6

Determine which combination of statements is true.
From the analysis: - Statement I is not necessarily true. - Statement II is true. - Statement III is true.
Thus, the combination of statements II and III is true.
The correct answer is (C) II and III only.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord