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Why is the sum of the reciprocals of all of the divisors of a perfect number equal to 2?
Suppose N is a perfect number. Then N cannot be a square number and so N has an even number of factors.Suppose the factors are f(1) =1, f(2), f(3), ... , f(k-1), f(k)=N.Furthermore f(r) * f(k+1-r) = N for r = 1, 2, ... k so that f(r) = N/f(k+1-r)which implies that 1/f(r) = f(k+1-r)/NThen 1/f(1) + 1/(f(2) + ... + 1/f(k)= f(k)/N + f(k-1)/N + ... + f(1)/N= [f(k) + f(k-1) + ... + f(1)] / N= 2N/N since, by definition, [f(k) + f(k-1) + ... + f(1)] = 2N
What is the rule to the pattern 1 3 7 15 31?
its Mersenne's Number f(n)= (2^n) -1 where n is the number of sequence. so f(1) = 1 f(2) = 3 f(3) = 7 f(4) = 15 f(5) = 31 ...
What describes the sequence 1 1 2 3 5 is it arithmetic or geometric?
It is an arithmetic sequence. To differentiate arithmetic from geometric sequences, take any three numbers within the sequence. If the middle number is the average of the two on either side then it is an arithmetic sequence. If the middle number squared is the product of the two on either side then it is a geometric sequence. The sequence 0, 1, 1, 2, 3, 5, 8 and so on is the Fibonacci series, which is an arithmetic sequence, where the next number in the series is the sum of the previous two numbers. Thus F(n) = F(n-1) + F(n-2). Note that the Fibonacci sequence always begins with the two numbers 0 and 1, never 1 and 1.
What is infinity divided by 5?
Infinity divided by any finite number is infinity. Here are the rules: 1. Infinity divided by a finite number is infinite (I / f = I); 2. Any finite number divided by infinity is a number infinitesimally larger than, but never equal to, zero (f / I = 1 / I); 3. Infinity divided by infinity is one (I / I = 1), or in fact any other positive number (I / I = and so on...); 4. Infinity multiplied by zero (no infinity) is zero (I * 0 = 0); 5. Infinity divided by a positive finite number is infinity (I / +f = I); 6. Infinity divided by a negative finite number is minus infinity (I / -f = -I); 7. Infinity divided by zero is not possible; 8. Infinity plus infinity is infinity (I + I = I); 9. Zero divided by infinity (nothing divided into infinity) equals zero (0 / I = 0); 10. Infinity plus a finite number is infinity (I + f = I); 11. Infinity minus a finite number is infinity (I - f = I); but 12. Infinity minus infinity, due to the nature of infinity, can be zero, infinity, or minus infinity (I - I = -I, 0, I).
What is oilers number?
You are referring to Euler's number. It is pronounced like "eh-uh-le-uh" because the name is German. Euler's number is the number e, which is the base of the natural logarithm. The number e is extremely prominent in any mathematical endeavour involving calculus in any way, as it has many useful properties. For example, the derivative of the function f(x) = e^x is the function df(x) = e^x = f(x). It is the only function that has this property. The natural logarithm is itself also very useful. It was discovered by studying properties of the integral of the function f(x) = 1/x, which did not have any closed form antiderivative in terms of the elementary functions of that time period, even though g(x) = x^n had the simple antiderivative Ig(x, C) = (x^(n+1))/(n+1) + C for any real n except n = -1. It was eventually discovered that the function f(x) defined as the definite integral of f(t) = 1/t from t = 1 to t=x had the properties of a logarithmic function with base e. e is transcendental, so it is both irrational and is not the root of any polynomial function with rational coefficients. Or in simple words 2.718281828459
