Math Problem Statement

Write the following expression as a single summation in terms of k: \(\sum_{k=1}^{m} \frac{k}{k+5} + \frac{m+1}{m+6}\)

Solution

To solve the problem, we are asked to express the given summation as a single summation in terms of kk.

The given expression is:

k=1mkk+5+m+1m+6\sum_{k=1}^{m} \frac{k}{k+5} + \frac{m+1}{m+6}

We need to combine these into a single summation. The first part is already a summation, but the second part is just a constant value m+1m+6\frac{m+1}{m+6}.

Step-by-step explanation:

  1. Separate the constant term: Notice that the second term, m+1m+6\frac{m+1}{m+6}, is not dependent on kk — it’s a constant with respect to kk. Therefore, it can be added to each term of the summation.

  2. Rewrite the sum: The constant m+1m+6\frac{m+1}{m+6} can be added inside the summation by summing it over the same limits: k=1mkk+5+m+1m+6=k=1m(kk+5)+k=1m(m+1m+6)\sum_{k=1}^{m} \frac{k}{k+5} + \frac{m+1}{m+6} = \sum_{k=1}^{m} \left( \frac{k}{k+5} \right) + \sum_{k=1}^{m} \left( \frac{m+1}{m+6} \right)

  3. Simplify the second summation: Since m+1m+6\frac{m+1}{m+6} is independent of kk, the second sum becomes: k=1mm+1m+6=mm+1m+6\sum_{k=1}^{m} \frac{m+1}{m+6} = m \cdot \frac{m+1}{m+6} because the same constant is being added mm times.

  4. Final expression: k=1mkk+5+m+1m+6=k=1mkk+5+mm+1m+6\sum_{k=1}^{m} \frac{k}{k+5} + \frac{m+1}{m+6} = \sum_{k=1}^{m} \frac{k}{k+5} + m \cdot \frac{m+1}{m+6}

This is the single summation expression you are asked to find.

Would you like more details or further clarification?

Here are 5 related questions to expand on this:

  1. How would this approach change if there was a different constant in the summation?
  2. How can we express the sum of constant terms more generally?
  3. How does the behavior of this summation change as mm increases?
  4. Can this summation be approximated or simplified further for large values of mm?
  5. How does this summation compare to other common summation patterns in calculus?

Tip: When working with sums that involve both variable and constant terms, always look for ways to factor the constant into the summation for simplification.

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Math Problem Analysis

Mathematical Concepts

Summation
Algebra
Series

Formulas

Summation of a series: \(\sum_{k=1}^{m} f(k)\)
Constant term handling in summations

Theorems

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Suitable Grade Level

College-level Mathematics or Advanced High School