I am teaching a course on basics statistics, and we are doing the t-test for two independent samples with unequal variances (Welch test). In the examples I have seen, the adjusted degrees of freedom used by the Welch test are always less than or equal to $n_1+n_2-2$.
Is this always the case? Does the Welch test always reduce (or leave unchanged) the degrees of freedom of the pooled (equal variances) t-test?
And on the same subject, if the sample standard deviations are equal, do the DF's of the Welch test reduce to $n_1+n_2-2$? I looked at the formula, but the algebra got messy.