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Telescoping series

Problem 7.493 · easy

Find the sum of \( \displaystyle \sum_{n=1}^{\infty} \frac{6}{n^{2} + 3 n + 2} \), or show that it diverges.

\[ \frac{6}{n^{2} + 3 n + 2} = - \frac{6}{n + 2} + \frac{6}{n + 1} \]
Partial fractions: the term is a difference.✓ Proved
In the N-th partial sum every negative piece cancels a positive piece 1 term later; only the first positive and last negative pieces survive.
So S_N = 3 - 6/(N + 2).
\[ \lim_{k \to \infty}\left(3 - \frac{6}{k + 2}\right) = 3 \]
Let N → ∞.✓ Proved

Answer \( 3 \)

Lines: 2 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
3	Not checked	—	a sentence; read, not computed
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	mpmath's nsum of the series (with convergence acceleration)

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The partial fraction decomposition in step 1 is incorrect; the correct decomposition is 6/(n+1) - 6/(n+2), not -6/(n+2) + 6/(n+1) (which is actually correct algebraically, wait. Let's re-read carefully. -6/(n+2) + 6/(n+1) IS 6/(n+1) - 6/(n+2). So step 1 is correct. Let's look at step 2 and 3. S_N = sum_{n=1}^N (6/(n+1) - 6/(n+2)). Terms: (6/2 - 6/3) + (6/3 - 6/4) + ... + (6/(N+1) - 6/(N+2)). Surviving terms: 6/2 - 6/(N+2) = 3 - 6/(N+2). Step 3 says S_N = 3 - 6/(N+2). This is correct. Step 4 limit is 3. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's instruction. 'Find what is wrong'. Is there a subtle error? Step 1: 6/(n^2+3n+2) = 6/((n+1)(n+2)) = 6(1/(n+1) - 1/(n+2)) = 6/(n+1) - 6/(n+2). The equation says -6/(n+2) + 6/(n+1). This is equivalent. Step 2 describes telescoping. Step 3 gives the partial sum. Step 4 takes the limit. Everything seems correct. Wait, let me check the partial fraction again. A/(n+1) + B/(n+2) = (A(n+2) + B(n+1))/(n+1)(n+2). A+B=0, 2A+B=6 => A=6, B=-6. So 6/(n+1) - 6/(n+2). The text says '-6/(n + 2) + 6/(n + 1)'. This is the same. The partial sum S_N = sum_{n=1}^N (6/(n+1) - 6/(n+2)). First term n=1: 6/2 - 6/3. Last term n=N: 6/(N+1) - 6/(N+2). Sum = 6/2 - 6/(N+2) = 3 - 6/(N+2). Limit is 3. The solution is correct. Is there a 'style' issue? Or is it a 'pass'? The prompt asks to find what is wrong. If nothing is wrong, I should say pass. But usually these adversarial reviews have a trap. Let's look really closely. 'In the N-th partial sum every negative piece cancels a positive piece 1 term later'. The negative piece of term n is -6/(n+2). The positive piece of term n+1 is 6/((n+1)+1) = 6/(n+2). They cancel. Correct. 'only the first positive and last negative pieces survive'. First positive is from n=1: 6/(1+1) = 3. Last negative is from n=N: -6/(N+2). Correct. S_N = 3 - 6/(N+2). Correct. Limit is 3. Correct. Okay, I will mark it as pass.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The partial fraction decomposition in step 1 is incorrect; the correct decomposition is 6/(n+1) - 6/(n+2), not -6/(n+2) + 6/(n+1) (which is actually correct algebraically, wait. Let's re-read carefully. -6/(n+2) + 6/(n+1) IS 6/(n+1) - 6/(n+2). So step 1 is correct. Let's look at step 2 and 3. S_N = sum_{n=1}^N (6/(n+1) - 6/(n+2)). Terms: (6/2 - 6/3) + (6/3 - 6/4) + ... + (6/(N+1) - 6/(N+2)). Surviving terms: 6/2 - 6/(N+2) = 3 - 6/(N+2). Step 3 says S_N = 3 - 6/(N+2). This is correct. Step 4 limit is 3. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's instruction. 'Find what is wrong'. Is there a subtle error? Step 1: 6/(n^2+3n+2) = 6/((n+1)(n+2)) = 6(1/(n+1) - 1/(n+2)) = 6/(n+1) - 6/(n+2). The equation says -6/(n+2) + 6/(n+1). This is equivalent. Step 2 describes telescoping. Step 3 gives the partial sum. Step 4 takes the limit. Everything seems correct. Wait, let me check the partial fraction again. A/(n+1) + B/(n+2) = (A(n+2) + B(n+1))/(n+1)(n+2). A+B=0, 2A+B=6 => A=6, B=-6. So 6/(n+1) - 6/(n+2). The text says '-6/(n + 2) + 6/(n + 1)'. This is the same. The partial sum S_N = sum_{n=1}^N (6/(n+1) - 6/(n+2)). First term n=1: 6/2 - 6/3. Last term n=N: 6/(N+1) - 6/(N+2). Sum = 6/2 - 6/(N+2) = 3 - 6/(N+2). Limit is 3. The solution is correct. Is there a 'style' issue? Or is it a 'pass'? The prompt asks to find what is wrong. If nothing is wrong, I should say pass. But usually these adversarial reviews have a trap. Let's look really closely. 'In the N-th partial sum every negative piece cancels a positive piece 1 term later'. The negative piece of term n is -6/(n+2). The positive piece of term n+1 is 6/((n+1)+1) = 6/(n+2). They cancel. Correct. 'only the first positive and last negative pieces survive'. First positive is from n=1: 6/(1+1) = 3. Last negative is from n=N: -6/(N+2). Correct. S_N = 3 - 6/(N+2). Correct. Limit is 3. Correct. Okay, I will mark it as pass.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The partial fraction decomposition in step 1 is incorrect; the correct decomposition is 6/(n+1) - 6/(n+2), not -6/(n+2) + 6/(n+1) (which implies the terms are swapped or signs are wrong relative to standard ordering, but more critically, the coefficients are correct but the cancellation logic in step 2 relies on the specific form. Actually, 6/(n^2+3n+2) = 6/((n+1)(n+2)) = 6/(n+1) - 6/(n+2). Step 1 writes -6/(n+2) + 6/(n+1), which is algebraically equivalent. However, Step 3 claims S_N = 3 - 6/(N+2). Let's check: S_N = sum_{n=1}^N (6/(n+1) - 6/(n+2)) = (6/2 - 6/3) + (6/3 - 6/4) + ... + (6/(N+1) - 6/(N+2)) = 6/2 - 6/(N+2) = 3 - 6/(N+2). The algebra is actually correct. Wait, let me re-read step 1. Step 1 says: 6/(n**2 + 3*n + 2) = -6/(n + 2) + 6/(n + 1). This is correct. Step 2 says negative cancels positive 1 term later. The term is A_n - A_{n+1} where A_n = 6/(n+1). So sum is A_1 - A_{N+1} = 6/2 - 6/(N+2) = 3 - 6/(N+2). Step 3 is correct. Step 4 limit is 3. The solution is actually correct. Why did I think it was wrong? Let me double check the partial fraction. 6/((n+1)(n+2)) = A/(n+1) + B/(n+2). A(n+2) + B(n+1) = 6. n=-1 => A(1) = 6 => A=6. n=-2 => B(-1) = 6 => B=-6. So 6/(n+1) - 6/(n+2). Step 1 writes -6/(n+2) + 6/(n+1). This is the same. The telescoping sum is correct. The limit is correct. The answer is 3. The solution is sound.

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/telescoping_series, checked 2026-10-10 with SymPy 1.14.0.