Find the sum 1/√(1+√2) + 1/√(2+√3) + ... + 1/√(n+√(n+1)).
Understand the Problem
The question is asking for the calculation of a mathematical sum that involves square roots and a sequence. The terms in the sum are of the form 1 over the square root of 'n' added to 'n + 1'. We need to evaluate this sum for the specified series.
Answer
The sum is $$ S = \sum_{k=1}^{n} \frac{\sqrt{k - \sqrt{k + 1}}}{\sqrt{k^2 - k - 1}} $$
Answer for screen readers
The sum is
$$ S = \sum_{k=1}^{n} \frac{\sqrt{k - \sqrt{k + 1}}}{\sqrt{k^2 - k - 1}} $$
Steps to Solve
- Identify the series structure
The sum can be expressed as:
$$ S = \sum_{k=1}^{n} \frac{1}{\sqrt{k + \sqrt{k + 1}}} $$
- Simplify each term in the series
We can simplify the term $\frac{1}{\sqrt{k + \sqrt{k + 1}}}$.
Start by multiplying the numerator and the denominator by $\sqrt{k - \sqrt{k + 1}}$ (the conjugate):
$$ \frac{1}{\sqrt{k + \sqrt{k + 1}}} \cdot \frac{\sqrt{k - \sqrt{k + 1}}}{\sqrt{k - \sqrt{k + 1}}} = \frac{\sqrt{k - \sqrt{k + 1}}}{\sqrt{(k + \sqrt{k + 1})(k - \sqrt{k + 1})}} $$
- Calculate the denominator
The denominator becomes:
$$ (k + \sqrt{k + 1})(k - \sqrt{k + 1}) = k^2 - (k + 1) = k^2 - k - 1 $$
Thus,
$$ S = \sum_{k=1}^{n} \frac{\sqrt{k - \sqrt{k + 1}}}{\sqrt{k^2 - k - 1}} $$
- Evaluate the sum analytically or numerically
This sum may not simplify nicely for all $n$, and numerical computation or further manipulation may be necessary to evaluate it fully.
The sum is
$$ S = \sum_{k=1}^{n} \frac{\sqrt{k - \sqrt{k + 1}}}{\sqrt{k^2 - k - 1}} $$
More Information
This sum represents a sequence with roots and requires advanced simplification techniques or numerical methods for specific evaluations.
Tips
- Forgetting to rationalize the denominator.
- Misidentifying the series limit.
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