No, because "will" and "should" are both modal verbs. You cannot use two modal verbs next to each other. You may say, "I will take part in this summer course!" or "I should take part in this summer course!"

Hello Saeed, "will" and "should" are called modal verbs. In the English grammar, you aren't allowed to put two modal verbs next to each other in a single sentence. So the best way is, to choose only one of them to make the sentence look correct!

Ariana is right. Both verbs are modal and it would be inappropriate to use both at the same time. "Should" implies a less than certian probability, whereas "will" implies an almost certain probability. It depends on how probable you are in taking the summer course.