Add telescoping square root sum calculator

[Copilot]

Implements calculation of the finite sum Σ 1/(√k + √(k+4)) for k = 1…n (specifically n = 2022), exploiting the telescoping structure after conjugate rationalisation.

Approach

Rationalising each term collapses the series:

1/(√k + √(k+4))  →  (√(k+4) − √k) / 4

Most intermediate terms cancel, leaving the O(1) closed form:

S = (√(n+1) + √(n+2) + √(n+3) + √(n+4) − 1 − √2 − √3 − 2) / 4

For n = 2022 → S ≈ 43.4578762937

New files

• calculate_sqrt_sum.py — two public functions:

  • sqrt_sum_analytical(n) — closed-form O(1) result via telescoping
  • sqrt_sum_direct(n) — brute-force term-by-term sum (used for cross-validation)
  • Both raise ValueError for n ≤ 0

• test_calculate_sqrt_sum.py — 12 unit tests covering single-term correctness, known closed-form value at n = 4, analytical/direct agreement at n = 100 and n = 2022, and invalid-input handling

Original prompt

How can I calculate
1
√
1
+
√
5
+
1
√
2
+
√
6
+
1
√
3
+
√
7
+
1
√
4
+
√
8
+
.
.
.
+
1
√
2021
+
√
2025
+
1
√
2022
+
√
2026
?
1
1
+
5
+
1
2
+
6
+
1
3
+
7
+
1
4
+
8
+
.
.
.
+
1
2021
+
2025
+
1
2022
+
2026
?

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