Дополнение к статье "Радикальна ознака Кошi"
t ⊢ a : N ↦ R ∧ ∀ n ∈ N ( a n > 0 ) ∧ ∃ d ∈ ( 0 , 1 ) ∃ N ∈ N ∀ n ∈ N ∧ n > N ( a n n < d ) → ( ∑ i = 0 ∞ a i ) ∈ R {\displaystyle ~t\vdash \ \mathrm {a} :\mathbb {N} \mapsto \mathbb {R} \ \ \land \ \ \forall _{n\ \in \ \mathbb {N} }\ (a_{n}>0)\ \ \land \ \ \exists _{d\ \in \ (0,1)}\ \exists _{N\ \in \ \mathbb {N} }\ \forall _{n\ \in \ \mathbb {N} \ \land \ n\ >\ N}\ ({\sqrt[{n}]{a_{n}}}<d)\to (\sum _{i=0}^{\infty }a_{i})\in \mathbb {R} }
t ⊢ a : N ↦ R ∧ ∀ n ∈ N ( a n > 0 ) ∧ lim n → ∞ a n n ∈ ( 0 , 1 ) → ( ∑ i = 0 ∞ a i ) ∈ R {\displaystyle ~t\vdash \ \mathrm {a} :\mathbb {N} \mapsto \mathbb {R} \quad \land \quad \forall _{n\ \in \ \mathbb {N} }\ (a_{n}>0)\quad \land \quad \lim _{n\to \infty }{\sqrt[{n}]{a_{n}}}\in (0,1)\quad \to \quad (\sum _{i=0}^{\infty }a_{i})\in \mathbb {R} }
t ⊢ a : N ↦ R ∧ ∀ n ∈ N ( a n > 0 ) ∧ lim n → ∞ a n n ∈ ( 1 , ∞ ) → lim n → ∞ ( ∑ i = 0 n a i ) = ∞ {\displaystyle ~t\vdash \ \mathrm {a} :\mathbb {N} \mapsto \mathbb {R} \quad \land \quad \forall _{n\ \in \ \mathbb {N} }\ (a_{n}>0)\quad \land \quad \lim _{n\to \infty }{\sqrt[{n}]{a_{n}}}\in (1,\infty )\quad \to \quad \lim _{n\to \infty }(\sum _{i=0}^{n}a_{i})=\infty }
Галактион 20:47, 15 серпня 2009 (UTC)Відповісти
![{\displaystyle ~t\vdash \ \mathrm {a} :\mathbb {N} \mapsto \mathbb {R} \ \ \land \ \ \forall _{n\ \in \ \mathbb {N} }\ (a_{n}>0)\ \ \land \ \ \exists _{d\ \in \ (0,1)}\ \exists _{N\ \in \ \mathbb {N} }\ \forall _{n\ \in \ \mathbb {N} \ \land \ n\ >\ N}\ ({\sqrt[{n}]{a_{n}}}<d)\to (\sum _{i=0}^{\infty }a_{i})\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c975d9f436837e925cf41724f31871c9dd08f5f8)
![{\displaystyle ~t\vdash \ \mathrm {a} :\mathbb {N} \mapsto \mathbb {R} \quad \land \quad \forall _{n\ \in \ \mathbb {N} }\ (a_{n}>0)\quad \land \quad \lim _{n\to \infty }{\sqrt[{n}]{a_{n}}}\in (0,1)\quad \to \quad (\sum _{i=0}^{\infty }a_{i})\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9faa2d25ad499815a95633b31ccf217208d81e2)
![{\displaystyle ~t\vdash \ \mathrm {a} :\mathbb {N} \mapsto \mathbb {R} \quad \land \quad \forall _{n\ \in \ \mathbb {N} }\ (a_{n}>0)\quad \land \quad \lim _{n\to \infty }{\sqrt[{n}]{a_{n}}}\in (1,\infty )\quad \to \quad \lim _{n\to \infty }(\sum _{i=0}^{n}a_{i})=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8e3b1607295cae52e3397ea83c4882e2602938)
![{\displaystyle ~(\lim _{n\to \infty }{\sqrt[{n}]{a_{n}}}\in (0,1)\ \to \ (\sum _{i=0}^{\infty }a_{i})\in \mathbb {R} )\quad \land \quad (\lim _{n\to n}{\sqrt[{n}]{a_{n}}}\in (1,\infty )\ \to \ \lim _{n\to \infty }(\sum _{i=0}^{n}a_{i})=\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8becc8ee31156a8d78f425f6eed4d3307b47e3c)
