Rewrite the sum using sigma notation. 1 1 1 1 + 1.2 2.3 3.4 4.5 Which summation represents the sum? ∞ 1 Ο Α.Σ k(k + 1) k=1 4 1 Ο Β. Σ k(k + 1) k=1 8 1 oc Σ k(k + 1) k=0 1 OD Σ –
Rewrite the sum using sigma notation. 1 1 1 1 + 1.2 2.3 3.4 4.5 Which summation represents the sum? ∞ 1 Ο Α.Σ k(k + 1) k=1 4 1 Ο Β. Σ k(k + 1) k=1 8 1 oc Σ k(k + 1) k=0 1 OD Σ –
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Rewriting Sums Using Sigma Notation
#### Problem Statement:
Rewrite the given sum using sigma notation:
\[ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \cdots \]
#### Question:
Which summation represents the given sum?
#### Options:
A.
\[ \sum_{k=1}^{\infty} \frac{1}{k(k+1)} \]
B.
\[ \sum_{k=1}^{4} \frac{1}{k(k+1)} \]
C.
\[ \sum_{k=0}^{\infty} \frac{1}{k(k+1)} \]
D.
\[ \sum_{l,1}^{8} \frac{1}{l} \]
In the given problem, you are asked to express the series
\[ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \cdots \]
using sigma (Σ) notation. The choices indicate different ways to represent this infinite sum. The correct answer should match exactly how the series is formed and indicate an infinite summation starting from an appropriate index value.
#### Details on provided options:
- **Option A:** The summation starts at \( k=1 \) and goes to infinity (\( \infty \)), and each term is given by the formula \(\frac{1}{k(k+1)} \).
- **Option B:** The summation starts at \( k=1 \) but only sums up to 4, which does not represent the infinite series.
- **Option C:** The summation starts at \( k=0 \) and goes to infinity (\( \infty \)), but given the formula, starting at \( k=0 \) will involve division by zero in the first term.
- **Option D:** Represents a different series altogether and does not fit the provided infinite series involving products of sequential integers.
The correct representation of the given series is:
\[ \sum_{k=1}^{\infty} \frac{1}{k(k+1)} \]
Therefore](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b6c3136-3b78-4c3b-95a7-be26ea6fb45d%2F057bb020-e8b3-4646-9a33-dbb6c48fef49%2Fnyxl08_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Rewriting Sums Using Sigma Notation
#### Problem Statement:
Rewrite the given sum using sigma notation:
\[ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \cdots \]
#### Question:
Which summation represents the given sum?
#### Options:
A.
\[ \sum_{k=1}^{\infty} \frac{1}{k(k+1)} \]
B.
\[ \sum_{k=1}^{4} \frac{1}{k(k+1)} \]
C.
\[ \sum_{k=0}^{\infty} \frac{1}{k(k+1)} \]
D.
\[ \sum_{l,1}^{8} \frac{1}{l} \]
In the given problem, you are asked to express the series
\[ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + \cdots \]
using sigma (Σ) notation. The choices indicate different ways to represent this infinite sum. The correct answer should match exactly how the series is formed and indicate an infinite summation starting from an appropriate index value.
#### Details on provided options:
- **Option A:** The summation starts at \( k=1 \) and goes to infinity (\( \infty \)), and each term is given by the formula \(\frac{1}{k(k+1)} \).
- **Option B:** The summation starts at \( k=1 \) but only sums up to 4, which does not represent the infinite series.
- **Option C:** The summation starts at \( k=0 \) and goes to infinity (\( \infty \)), but given the formula, starting at \( k=0 \) will involve division by zero in the first term.
- **Option D:** Represents a different series altogether and does not fit the provided infinite series involving products of sequential integers.
The correct representation of the given series is:
\[ \sum_{k=1}^{\infty} \frac{1}{k(k+1)} \]
Therefore
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