B What is a closed form solution?

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A closed form solution is an expression that can be computed in a finite number of standard operations, such as addition, multiplication, and exponentiation. An example of a closed form solution is the finite sum $$\sum_{k=1}^5 \frac{1}{2^k}$$, which can be evaluated directly. In contrast, the infinite series $$\sum_{k=1}^\infty \frac{1}{2^k}$$ converges to a limit but is not considered a closed form solution because it requires an understanding of limits and convergence. Closed form solutions are often preferred in mathematics for their simplicity and ease of use. Understanding the distinction between closed form and open form solutions is essential for mathematical problem-solving.
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What is the definition of a closed form solution in math? Where did the term originate? When is it preferred?
Hi friends, I was wondering if you could give the definition of 'closed form', with examples of closed form solutions and open? form solutions.

Foe example, is this a closed form solution?
$$\sum_{k=1}^\infty \frac{1}{2^k}$$

Or this?
$$\sum_{k=1}^5 \frac{1}{2^k}$$

Thanks.
 
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Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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