What does the $\prod$ symbol mean? - Mathematics Stack Exchange 21 The symbol $\Pi$ is the pi-product It is like the summation symbol $\sum$ but rather than addition its operation is multiplication For example, $$ \prod_ {i=1}^5i=1\cdot2\cdot3\cdot4\cdot5=120 $$ The other symbol is the coproduct
Finding the limit $\lim_ {x \to 0} \frac {1-\prod_ {i=1}^n\cos^ {1 i . . . By L'Hospital: The derivative of the denominator is (by pulling one cosine at a time from the product) $$\sum_ {i=1}^n\frac {i\sin (ix)} {\cos (ix)}\prod_ {i=1}^n\cos (ix) $$ This still tends to $0$ so we differentiate a second time and get $$\sum_ {i=1}^n\frac {i^2} {\cos^2 (ix)}\prod_ {i=1}^n\cos (ix)-\left (\sum_ {i=1}^n\frac {i\sin (ix)} {\cos (ix)}\right)^2\prod_ {i=1}^n\cos (ix),$$ which
elementary number theory - Mathematics Stack Exchange There are at least $p_n- 1$ primes between $p_n$ and $\prod_ {k=1}^n p_k$ · This is an exercise in Władysław Narkiewicz's book The Development of Prime Number Theory